Find x if log_2 x^2 + log_1/2 x = 5.

log₂ x^2 + log _1/2 x   =  5

2 log₂ 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

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