#1**+1 **

log_{2} x^2 + log _{1/2} x = 5

2 log_{2} x + log _{1/2} x = 5 use the change-of-base rule to write

2 [ log x / log 2] + log x / log (1/2) = 5 { log (1/2) = log 2^(-1) }

2[log x ] / log 2 ] + log x / log 2^(-1) = 5

2[;og x / log 2 ] + log x / -log 2 = 5

2 [ log x / log 2 ] - logx/ log 2 = 5

[2logx - log x ] / log 2 = 5

log x/ log 2 = 5

log x = 5log 2

log x = log 2^5

log x = log 32

x = 32

CPhill Feb 22, 2018

#2**0 **

Solve for x:

(log(x^2))/log(2) - log(x)/log(2) = 5

Rewrite the left hand side by combining fractions. (log(x^2))/log(2) - log(x)/log(2) = (log(x^2) - log(x))/log(2):

(log(x^2) - log(x))/log(2) = 5

Multiply both sides by log(2):

log(x^2) - log(x) = 5 log(2)

log(x^2) - log(x) = log(1/x) + log(x^2) = log(x^2/x) = log(x):

log(x) = 5 log(2)

5 log(2) = log(2^5) = log(32):

log(x) = log(32)

Cancel logarithms by taking exp of both sides:

**x = 32**

Guest Feb 22, 2018