Math  /  Algebra

Question3log2xlog2x=83 \log _{2} x-\log _{2} x=8

Studdy Solution

STEP 1

1. The equation 3log2xlog2x=8 3 \log_{2} x - \log_{2} x = 8 involves logarithmic expressions.
2. We will use properties of logarithms to simplify and solve for x x .

STEP 2

1. Simplify the logarithmic expression.
2. Solve the resulting equation for x x .

STEP 3

First, simplify the expression 3log2xlog2x 3 \log_{2} x - \log_{2} x . This can be done by combining like terms:
3log2xlog2x=(31)log2x 3 \log_{2} x - \log_{2} x = (3 - 1) \log_{2} x 2log2x 2 \log_{2} x

STEP 4

Now, set the simplified expression equal to 8:
2log2x=8 2 \log_{2} x = 8
Divide both sides by 2 to isolate the logarithm:
log2x=82 \log_{2} x = \frac{8}{2} log2x=4 \log_{2} x = 4

STEP 5

To solve for x x , rewrite the logarithmic equation in its exponential form. Recall that if logba=c \log_{b} a = c , then a=bc a = b^c :
x=24 x = 2^4
Calculate the power:
x=16 x = 16
The value of x x is:
16 \boxed{16}

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