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Solve the logarithmic equation. Round to the nearest ten-thousandth if necessary.
3 log 2x = 4
To solve the logarithmic equation 3 log 2x = 4, we need to isolate the logarithm.
First, we can divide both sides of the equation by 3:
log 2x = 4 / 3
Next, we can rewrite the equation in exponential form:
2x = 10^(4/3)
To simplify the equation further, we can convert 10^(4/3) into a decimal approximation:
10^(4/3) ≈ 10^1.3333 ≈ 19.9526
Therefore, the equation becomes:
2x = 19.9526
To solve for x, we need to take the logarithm of both sides. Let's take the natural logarithm (ln) since the base of 2 in the original equation is not specified:
ln(2x) = ln(19.9526)
Using the property of logarithms, ln(2x) = x ln(2):
x ln(2) = ln(19.9526)
Now, we can divide both sides of the equation by ln(2):
x ≈ ln(19.9526) / ln(2)
Using a calculator, we can find the approximate value of ln(19.9526) / ln(2):
x ≈ 4.9617
Therefore, the solution to the logarithmic equation 3 log 2x = 4 is approximately x ≈ 4.9617.