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. Take the natural logarithm of both sides of the equation: $\ln (8^x)=\ln (12143)$.
2. Use the power rule of logarithms to bring down the exponent: $x\cdot \ln (8)=\ln (12143)$.
3. Solve for $x$ by dividing both sides by $\ln (8)$: $x=\frac{\ln (12143)}{\ln (8)}$.
4. Use a calculator to find the values of $\ln (12143)$ and $\ln (8)$.
5. Divide the two logarithmic values to find the decimal approximation of $x$, rounding to two decimal places.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions in which a variable appears in the exponent. To solve these equations, one typically seeks to isolate the variable by using properties of exponents or logarithms. For example, in the equation 8^x = 12143, the goal is to express x in a form that can be calculated, often by taking the logarithm of both sides.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for exponents in equations. The logarithm of a number is the exponent to which a base must be raised to produce that number. In the context of the equation 8^x = 12143, one would use logarithms to rewrite the equation as x = log_8(12143), which can be further simplified using the change of base formula.

Calculator Use for Approximations

Using a calculator to obtain decimal approximations is essential when dealing with logarithmic values that do not yield simple fractions. After determining the logarithmic expression for x, a calculator can provide a numerical approximation, which is often required in practical applications. For instance, after calculating log_8(12143), one would use a calculator to find the decimal value of x, rounding it to two decimal places as specified.