Question

Solve the equations. $$ \left(\frac{2}{3}\right)^{y}=\frac{5}{7} $$

   Solve the equations.
$$
\left(\frac{2}{3}\right)^{y}=\frac{5}{7}
$$

Algebra Form and Function

Algebra Form and Function

William G. McCallum,… 1st Edition

Chapter 11, Problem 9

Instant Answer

Solved by Expert Nick Johnson

04/07/2022

Step 1

We use the base 10 logarithm (common logarithm) for simplicity. This gives us: $$ \log\left(\left(\frac{2}{3}\right)^{y}\right)=\log\left(\frac{5}{7}\right) $$ Show more…

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Solve the equations. $$ \left(\frac{2}{3}\right)^{y}=\frac{5}{7} $$

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Key Concepts

Exponential Equations

Exponential equations are mathematical statements in which variables appear as exponents. Solving these equations typically involves rewriting the equation in logarithmic form or applying logarithmic operations to isolate the variable in the exponent.

Logarithms

Logarithms are the inverse operations of exponentiation and serve as a powerful tool for solving exponential equations. By taking the logarithm of both sides of an equation, you can leverage properties of logarithms to linearize the exponent, making it possible to isolate and solve for the variable.

Properties of Logarithms

Key properties of logarithms include the power rule (log(a^b) = b log(a)) and the change of base formula. The power rule allows you to bring an exponent in front of the logarithm, facilitating the process of solving for that exponent, while the change of base formula provides flexibility in choosing the logarithm base to simplify the computations.

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