Problem 60 - 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. $$\log _{2}(4 x+1)=5$$

Step-by-Step Solution

Answer

The exact solution to the problem is \(x = 7.75\), which is also the decimal approximation up to two decimal places.

Step 1: Apply the Logarithm Properties

According to the properties of logarithms, the equation \(\log _{2}(4 x+1)=5\) can be rewritten as \(2^5 = 4x + 1\). This process is known as exponentiation.

Step 2: Solve for \(x\)

Having the equation as \(2^5 = 4x + 1\), simplify the left side to obtain \(32 = 4x + 1\). By isolating \(x\) on one side of the equation, subtract 1 from both sides to be left with \(31 = 4x\). Finally, divide both sides by 4 to obtain \(x = 31/4 = 7.75\).

Step 3: Check the Validity of the Solution

Substituting \(x = 7.75\) into the original logarithmic function to check its validity. If substituting \(x\) into the the original logarithmic function yields a true statement and the number falls in the domain of the original logarithmic function, then it is a valid solution. The function, \(\log _{2}(4 x+1)\), is defined for \(4x+1 > 0\), hence \(x > -1/4\). As \(x = 7.75 > -1/4\), the solution is valid.

Step 4: Decimal Approximation

For decimal approximation, convert the fraction into a decimal number up to two decimal places. Here, the fraction is already in decimal form, so \(x \approx 7.75\).