Solve exponential equations using common logarithms

Learning Objectives Identify an exponential equation that requires logarithms for its solution. Explain why taking the common logarithm of both sides of an equation is a valid step. Apply the Power Property of Logarithms to bring a variable exponent down as a coefficient. Solve for a variable in an exponential equation by isolating it after applying logarithms. Use a calculator to find the approximate decimal value of a logarithmic expression. Verify the solution to an exponential equation by substituting it back into the original equation. How long would it take for a $1,000 investment to grow to $5,000 at an 8% annual interest rate? 📈 Exponential equations hold the key, and logarithms are the tool to unlock the answer! In this tutorial, you will learn a powerful techniqu...

TermDefinitionExample Exponential EquationAn equation where the variable appears in the exponent.3^x = 81 or 5^(2x-1) = 50 LogarithmA logarithm is the exponent to which a base must be raised to produce a given number. It's the inverse operation of exponentiation.log₂(8) = 3, because 2³ = 8. Common LogarithmA logarithm with a base of 10. It is usually written as 'log(x)' without a visible base.log(100) = 2, because 10² = 100. Base (of an exponent)The number that is being raised to a power in an exponential expression.In 7⁴, the base is 7. Power Property of LogarithmsThis property allows you to move an exponent from inside a logarithm to the front as a coefficient.log(5³) can be rewritten as 3 * log(5).

Equality Property of Logarithms If a = b, then \log(a) = \log(b), for a > 0 and b > 0. This rule allows us to take the common logarithm of both sides of an equation without changing the equality. This is the foundational step for solving exponential equations. Power Property of Logarithms \log_b(M^p) = p \cdot \log_b(M) This is the most critical rule for this topic. It allows us to take a variable that is stuck in an exponent and move it to the front as a coefficient, turning an exponential problem into a linear one.