Math III
Go further with polynomial and rational expressions, advanced functions, trigonometry, geometric modeling, and statistical inference.
- Problem types
- 641
- Practice variants
- 2,564
Page 11 of 18
Each problem type has four distinct practice variants. Open a preview to move among all four.
Solve exponential equations by rewriting both sides with a common base and setting the exponents equal
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Before reaching for logarithms, check whether the constant side is an exact power of the exponential base. Rewriting both sides with one valid base lets the one-to-one property turn the …
Preview problemSolve exponential equations of the form \(ae^(ct)=d\) by isolating the exponential factor and using natural logarithms
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Natural logarithms are the direct inverse tool for an exponential with base e. Apply the logarithm to both sides so the exponent becomes an ordinary linear expression, then divide by …
Preview problemSolve exponential equations of the form \(a(b)^(ct)=d\) by isolating the exponential factor and taking logarithms
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Use a logarithm that matches the exponential base to expose the exponent in one step. After the inverse operations cancel, solve the resulting linear equation by dividing the entire logarithm …
Preview problemSolve an exponential equation of the form ab^(ct)=d by isolating the exponential factor and taking logarithms
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Turn the target amount into a target growth factor by dividing out the initial amount first. Logarithms then bring the unknown time down from the exponent, producing a quotient of …
Preview problemSolve an exponential equation by isolating the exponential factor and taking logarithms
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
First express the target as a fraction of the initial amount. In a half-life model, rewrite that fraction as a power of one half; matching bases then tells how many …
Preview problemEvaluate a logarithm by change of base to four decimals
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Change of base keeps the original argument in the numerator and the original base in the denominator, using the same calculator logarithm in both places. Evaluate the quotient at full …
Preview problemEvaluate a logarithm using change of base
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
A logarithm asks for the exponent that makes its base produce the argument. Change of base preserves that meaning by dividing the logarithm of the argument by the logarithm of …
Preview problemSolve and interpret an exponential threshold time
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Separate the continuous crossing time from the first completed reporting period. Cancel the positive initial amount, solve the resulting growth-factor equation with logarithms, and keep its decimal as the exact …
Preview problemSeparate real existence from contextual validity
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Check existence before applying logarithms. Once the exponential is isolated, compare its target with the range of a positive-base exponential, which is always positive for real inputs. Only a real …
Preview problemSolve an exponential equation exactly and to four decimals
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Taking logarithms turns the unknown exponent into a coefficient, leaving a linear equation whose solution is an exact logarithmic quotient. Keep that quotient intact for symbolic substitution, and use a …
Preview problemSolve an exponential inequality by isolating the exponential factor and using logarithms
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Treat the related equation as the boundary, then use monotonicity to determine which side of that boundary satisfies the inequality. Taking logarithms preserves order, but the sign of the logarithm …
Preview problemSolve an exponential equation by isolating the exponential factor and taking logarithms
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Method comes before machinery here. Check whether the constant is an exact power of the base already present; if it is, rewrite both sides with that base and use one-to-one …
Preview problemExpand a logarithm of a product without losing its domain
Prove simple logarithm laws.
Separate only genuine multiplicative factors inside the logarithm; the product law turns those factors into a sum of logs. Before rewriting, solve where the original argument is positive, then confirm …
Preview problemExpand a logarithm of a quotient and preserve its maximal domain
Prove simple logarithm laws.
For a logarithm of a quotient, first require the entire quotient to be positive and its denominator to be nonzero. On that domain, the quotient law becomes the numerator's logarithm …
Preview problemUse the logarithm power law without shrinking the real domain
Prove simple logarithm laws.
The power law moves an exponent outside as a multiplier, but domain analysis comes first. Determine when the original powered argument is positive, paying attention to whether the power preserves …
Preview problemCondense a sum or difference of logarithms into one logarithm
Prove simple logarithm laws.
Condensing reverses the logarithm laws: a sum of same-base logarithms becomes the logarithm of a product, while a difference would become a quotient. Check every original log argument first and …
Preview problemFind domain restrictions when applying logarithm laws
Prove simple logarithm laws.
Each logarithm contributes its own strict positivity condition. Solve those inequalities separately and intersect them, since every log must exist at the same input. This original common domain remains part …
Preview problemAudit a logarithm rewrite for equality and domain equivalence
Prove simple logarithm laws.
Audit equality and domain equivalence as separate questions. A positive product can come from two positive factors or two negative factors, while separate real logarithms without absolute values cover only …
Preview problemSolve simple logarithmic equations by combining logs and rewriting in exponential form
Prove simple logarithm laws.
Record the original log domain before changing the equation. Condense the sum into one logarithm, use one-to-one behavior to equate its positive argument with the other side's argument, and solve …
Preview problemDerive the logarithm product law from exponents
Prove simple logarithm laws.
The product law is the equal-base exponent rule viewed through inverse functions. Translate each logarithm statement into an exponential equation, multiply those equations, and add the exponents on the shared …
Preview problemDerive and domain-check the logarithm power law
Prove simple logarithm laws.
Derive the power law by translating the logarithm into an exponential statement. Raising that statement to a real power invokes the power-of-a-power rule, which multiplies the exponents; translating back produces …
Preview problemRewrite a logarithmic equation in exponential form
Use the definition of logarithms to translate among logarithms in any base.
A logarithmic statement assigns three fixed roles. The subscript remains the exponential base, the value of the logarithm becomes the exponent, and the argument becomes the resulting power. Reading the …
Preview problemRewrite an exponential equation in logarithmic form
Use the definition of logarithms to translate among logarithms in any base.
When moving from exponential to logarithmic form, preserve the original base. The power's result becomes the logarithm's argument, and the original exponent becomes the logarithm's value. A quick verbal check …
Preview problemEvaluate a logarithm by rewriting it in exponential form
Use the definition of logarithms to translate among logarithms in any base.
Treat the logarithm's value as an unknown exponent on its base. Rewrite the argument as an exact power of that same base, then use one-to-one behavior to match the exponents. …
Preview problemEvaluate an exact logarithm by rewriting it as an exponential equation
Use the definition of logarithms to translate among logarithms in any base.
Exact logarithms are exponent questions, so look for a power relationship before using a calculator. Translate the logarithm into a base-to-an-unknown-power equation and factor the argument into repeated copies of …
Preview problemSolve a logarithmic equation by rewriting it in exponential form
Use the definition of logarithms to translate among logarithms in any base.
Use the logarithm definition to move the unknown argument into an exponential equation. The log value becomes the exponent on the stated base, so evaluating that power isolates the unknown …
Preview problemSolve an exponential equation exactly by rewriting it in logarithmic form
Use the definition of logarithms to translate among logarithms in any base.
An exponential equation with no convenient common base can still name its unknown exponent exactly. Rewrite that exponent as a logarithm with the original base, then apply change of base …
Preview problemInterpret a logarithm as a closed multiplicative-interval model
Use the definition of logarithms to translate among logarithms in any base.
Translate the logarithm into a repeated-growth model by assigning each part a contextual role. The base is the multiplier per equal interval, the argument is the final-to-initial target ratio, and …
Preview problemDetermine the conditions that make a logarithm defined
Use the definition of logarithms to translate among logarithms in any base.
Check the argument and base independently. A real logarithm needs a positive argument, while its base must support a positive, one-to-one exponential inverse: it must be positive but cannot be …
Preview problemCompare logarithms in different bases using the definition of a logarithm
Use the definition of logarithms to translate among logarithms in any base.
A shared argument does not make logarithms equal when their bases differ. Translate each one into the exponent required for its own base to reach that argument, evaluate those powers …
Preview problemRewrite a logarithm in a calculator-friendly base using change of base
Use the definition of logarithms to translate among logarithms in any base.
Change of base keeps the logarithm’s argument in the numerator and its original base in the denominator, using the same calculator logarithm above and below. Check that order against neighboring …
Preview problemTranslate between logarithmic form and exponential form
Use the definition of logarithms to translate among logarithms in any base.
Translation changes the notation, not the roles: the subscript remains the power base, the logarithm value becomes the exponent, and the argument is the result. Reading the statement aloud as …
Preview problemCombine a sum of logarithms using the product law
Use logarithm properties to simplify numeric logarithmic expressions and estimate values.
When same-base logarithms are added, think multiplication inside a single logarithm rather than addition of the arguments. After combining, look for an exact power of the shared base; that turns …
Preview problemSimplify a difference of logarithms using the quotient law
Use logarithm properties to simplify numeric logarithmic expressions and estimate values.
A difference of logarithms with the same base packages the first argument divided by the second into one logarithm. Keep their order intact, simplify that quotient, and then ask which …
Preview problemSimplify logarithmic expressions using the power law
Use logarithm properties to simplify numeric logarithmic expressions and estimate values.
The power law moves a multiplier outside a logarithm onto the argument as an exponent; it does not multiply the argument or alter the base. Once the powered argument is …
Preview problemCondense a sum, difference, and coefficient of logarithms into a single logarithm
Use logarithm properties to simplify numeric logarithmic expressions and estimate values.
Condensing several same-base logarithms is an operation map: added terms contribute factors to the numerator, while subtracted terms contribute factors to the denominator. Build that single argument before doing arithmetic, …
Preview problem