Question

solve the logarithmic equation. round to the nearest ten - thousandth if necessary. 80. $3\log 2x = 4$ a. 10.7722 b. 5 c. 2.7826 d. 0.6309

Explanation:

Step1: Isolate the logarithmic term

Divide both sides of the equation \(3\log 2x = 4\) by 3.
\(\log 2x=\frac{4}{3}\)

Step2: Convert to exponential form

Recall that if \(\log_{10}a = b\), then \(a = 10^{b}\). Here, the base of the logarithm is 10 (common logarithm), so we can rewrite the equation as:
\(2x = 10^{\frac{4}{3}}\)

Step3: Calculate \(10^{\frac{4}{3}}\)

First, \(\frac{4}{3}\approx1.3333\), and \(10^{1.3333}\approx21.5443\) (using a calculator to find the value of \(10^{\frac{4}{3}}\)).

Step4: Solve for \(x\)

Divide both sides of the equation \(2x = 21.5443\) by 2:
\(x=\frac{21.5443}{2}= 10.77215\approx10.7722\)

Answer:

a. 10.7722