Problem 86 - College Algebra Essentials

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. $$\log (x-2)+\log 5=\log 100$$

Step-by-Step Solution

Answer

The solution to the given logarithmic equation is \( x = 22 \)

Step 1: Consolidate the Logarithms on Left Hand Side

Using the property of logarithms that states \( \log A + \log B = \log AB\), we can rewrite the equation as \( \log ( (x - 2) * 5) = \log 100 \). Simplifying further, we obtain \( \log (5x - 10) = \log 100 \).

Step 2: Cancel the Logs

Once we've obtained logarithms of two quantities on either side of the equation, we can drop the logs base 10. This will give us the equation \( 5x - 10 = 100 \).

Step 3: Solve for X

Solving the equation \( 5x - 10 = 100 \) for \( x \), first add 10 to both sides: \( 5x = 110 \). Next, divide both sides by 5 to solve for \( x \), yielding \( x = 22 \).

Step 4: Check the Solution

We've found the solution \( x = 22 \), but we need to check if it lies within the domain of the original logarithmic expressions. Substituting \( x = 22 \) into \( \log (x - 2) \), we get \( \log (22 - 2) \), which is valid as log is defined for all positive values. Therefore, \( x = 22 \) is indeed the solution.

Step 5: Decimal Approximation

The exact solution is \( x = 22 \). In this case, there is no need for a decimal approximation, since the solution is already a precise number.