Problem 23 - College Algebra Essentials

Question

Solve each exponential equation. 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. $$10^{x}=3.91$$

Step-by-Step Solution

Answer

The solution to the equation \(10^{x}=3.91\) in terms of a logarithm is \(x = log_{10} (3.91)\). The decimal approximation of this value, correct to two decimal places, can be calculated using a calculator.

Step 1 - Rewrite the equation in logarithmic form

We can rewrite the equation using the property of logarithms that allows us to switch between exponential and logarithmic form. That is, converting the equation from the form \(b^{x} = y\) to the form \(x = log_{b} (y)\), so \(10^{x} = 3.91\) becomes \(x = log_{10} (3.91)\). This allows to directly see the solution in logarithmic form.

Step 2 - Calculation in logarithmic form

The next step will be to calculate our logarithm with base 10 (log) in terms of the operations that the logarithm allows, i.e. multiplication, division, etc. In the given case, we only need to calculate the value of \(log_{10} (3.91)\). Take a quick note that log without a specified base is assumed to be in base 10.

Step 3 - Compute the decimal approximation

To obtain the decimal approximation with two decimal places, a calculator should be used to calculate the value of \(log_{10} (3.91)\).