Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 10^x=3.91

Verified step by step guidance

1. Identify the given exponential equation: \$10^{x} = 3.91$.
2. To solve for \(x\), take the logarithm of both sides. You can use either the common logarithm (base 10) or the natural logarithm (base \(e\)). For example, applying the common logarithm gives: \(\log(10^{x}) = \log(3.91)\).
3. Use the logarithmic identity \(\log(a^{b}) = b \log(a)\) to simplify the left side: \(x \log(10) = \log(3.91)\).
4. Since \(\log(10) = 1\), the equation simplifies to \(x = \log(3.91)\).
5. To find the decimal approximation, use a calculator to evaluate \(\log(3.91)\) and round the result to two decimal places.

Key Concepts

Exponential Equations

An exponential equation is one in which the variable appears in the exponent, such as 10^x = 3.91. Solving these equations often requires rewriting or applying logarithms to isolate the variable and find its value.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials, allowing us to solve for variables in exponents. Common logarithms (base 10) and natural logarithms (base e) are used to rewrite equations like 10^x = 3.91 as x = log(3.91) or x = ln(3.91)/ln(10).

Using Calculators for Approximation

After expressing the solution in logarithmic form, calculators help find decimal approximations. This step involves evaluating logarithmic expressions and rounding the result to a specified precision, such as two decimal places.