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 equation: ${10}^x=3.91$.
2. Take the common logarithm (base 10) of both sides: $\log ({10}^x)=\log (3.91)$.
3. Apply the power rule of logarithms: $x\cdot \log (10)=\log (3.91)$.
4. Since $\log (10)=1$, simplify the equation to $x=\log (3.91)$.
5. Use a calculator to find the decimal approximation of $\log (3.91)$ to two decimal places.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a variable appears in the exponent. To solve these equations, one typically uses logarithms, which are the inverse operations of exponentiation. For example, in the equation 10^x = 3.91, we can apply logarithms to isolate the variable x.

Logarithms

Logarithms are a way to express exponents in a different form. The logarithm of a number is the exponent to which a base must be raised to produce that number. In this case, using common logarithms (base 10), we can rewrite the equation as x = log(3.91), allowing us to solve for x.

Calculator Use for Approximations

Using a calculator to find decimal approximations of logarithmic values is essential for practical applications. After determining the logarithmic expression, inputting it into a scientific calculator provides a numerical solution. For instance, calculating log(3.91) will yield a decimal value, which can be rounded to two decimal places for clarity.