Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{5} x=3$$

Step-by-Step Solution

Answer

The exact answer is \(x = 125\), and the decimal approximation is 125.00.

Step 1: Convert the Logarithmic Equation to an Exponential Equation

Our original equation is \(\log_5 x = 3\). To change this logarithmic equation into an exponential one, one must use the property of logarithms that states \(\log_b a = n\) translates to \(b^n = a\). For this equation, using this property would give \(5^3 = x\).

Step 2: Calculate the Value of X

Now that we have the equation \(5^3 = x\), we can calculate the value of \(x\).\(5^3\) equals 125, so \(x = 125\). Since the logarithm of a negative number is undefined and 125 is a positive number, we don't have to reject this value.

Step 3: Give the Decimal Approximation

The exact answer is \(x = 125\). Since this value has no fractions or roots, it doesn't need a decimal approximation. But for completeness sake, the decimal value rounding to two decimal places would also be 125.00.