In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $8^x=12143$

Verified step by step guidance

1. Start with the given exponential equation: \$8^x = 12143$.
2. To solve for \(x\), take the natural logarithm (or common logarithm) of both sides to utilize the logarithm property that allows exponents to be brought down: \(\ln(8^x) = \ln(12143)\).
3. Apply the logarithm power rule: \(x \cdot \ln(8) = \ln(12143)\).
4. Isolate \(x\) by dividing both sides of the equation by \(\ln(8)\): \(x = \frac{\ln(12143)}{\ln(8)}\).
5. Use a calculator to evaluate the logarithms and compute the decimal approximation of \(x\), rounding your answer to two decimal places.

Key Concepts

Exponential Equations

An exponential equation is one in which the variable appears in the exponent. Solving such equations often involves rewriting the equation to isolate the exponential expression and then applying logarithms to solve for the variable.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentiation. They allow us to solve equations where the variable is an exponent by converting the exponential form into a logarithmic form, making it easier to isolate and solve for the variable.

Using Calculators for Approximations

Calculators can evaluate logarithms and provide decimal approximations of solutions. After expressing the solution in logarithmic form, a calculator helps find a numerical value, often rounded to a specified number of decimal places for practical use.