Problem 86 - 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. $$\log (x-2)+\log 5=\log 100$$

Step-by-Step Solution

Answer

The exact solution to the equation is \(x = 22\).

Step 1: Identify properties of logarithms

Recall the property of logarithms that the sum of two logarithms (with the same base) is equal to the logarithm of the product of the two quantities, in other words, \(\log (a)+\log (b)=\log (a.b)\).

Step 2: Simplify the left-hand side

Apply this property to the left-hand side of the equation: \(\log (x-2)+\log 5= \log ((x-2)5)\) which simplifies to \(\log (5x-10)\). So our equation becomes \(\log (5x-10)=\log 100\).

Step 3: Equate the arguments

Since the two expressions share the same base log, we can set their arguments equal to each other: \(5x - 10 = 100\).

Step 4: Solve for \(x\)

Solve the resulting equation for \(x\). Add 10 to both sides and divide by 5 gives: \(x = \frac{110}{5} = 22\).

Step 5: Check domain restrictions

Check if the obtained value in the original logarithmic expressions doesn't violate the domain, i.e., for \(\log (x-2)\), \(x\) must be larger than 2, otherwise, the value is undefined. Considering \(x = 22\), this condition is fulfilled.

Step 6: Decimal approximation

As we already got an integer, there is no need for decimal approximation in this case. However, if the result was not an integer, it could be rounded to two decimal places using a calculator.