Math help ASAP: logs?

Question

if you could at least do one example of each type (A and B), it would be really helpful. Thanks! i just need help figuring out the method of base changing.

solve for x by changing to the appropriate base

A. log(subscript 2)x=log(sub 4)25

log(sub 3)x=log(sub 9)7x-6

log(sub 2)x+log(sub 8)32x=1

solve each equation by changing the base of the logarithm to 10

B. log(sub 2)x=log(sub 5)3

log(sub 7)x=log(sub 2)9

log(sub 2)x+log(sub 4)x=log(sub 2)5

log(sub 3)x=5log(sub 10)2

Answer 1

First, I may want to point out that the "subscript" should read "base".

* The solution is long. Don't just copy the lines. Check every step in case of typo mistake

Note that: log (base a) b = log(base c)b / log(base c)a

c is a positive rational number

Now for A:

log (base2) x = log (base4) 25

log(base2) x / log(base2) 2 = log(base2) 25 / log(base2) 4

log(base2) x / 1  = log (base2) 25 / 2

note that: a log(base c) b = log(base c) b^a

log (base2) 25 / 2 = 0.5 log (base2) 25

= log (base2) 25^0.5 = log (base2) 5

therefore, log(base2) x = log (base2) 5

x = 5

You may do the rest by the same way.

log(base3)x = log(base9)(7x - 6)

log (base3) x / log(base3) 3 = log(base3)(7x-6) / log(base3)9

log(base3) x / 1 = log(base3) (7x-6) / 2

2log(base3) x = log (base3) (7x-6)

log(base3) (x^2) = log(base3) (7x-6)

x^2 = 7x-6

x^2 - 7x + 6 = 0

x = 1 or 6

log(base2)x+log(base8)32x=1

log(base2)x / log(base2)2 + log(base2)32x / log(base2)8 = 1

log(base2)x / 1 + log(base2)32x / 3 = 1

3log(base2)x + log(base2)32x = 3

log(base2)(x^3) + log(base2)32x = 3

note that log(base c) a + log(base c) b = log(base c) ab

log(base2)(x^3 * 32x) = 3 = log(base2)(2^3)=log(base2)8

32x^4 = 8

x^4 = 8/32 = 1/4

x = 1/(square-root 2) = 0.7071

Part B.

log(base2)x=log(base5)3

log x / log 2 = log 3 / log 5

log x = log 3 * log 2 / log 5

log x = 0.205485 = log (10^0.205485)

x = 10^0.205485 = 1.6050

log(base7)x=log(base2)9

log x / log 7 = log 9 / log 2

log x = log 9 * log 7 / log2

log x = 2.67890 = log (10^2.67890)

x = 10^2.67890 = 477.416

log(base2)x+log(base4)x=log(base2)5

log x / log 2 + log x / log 4 = log 5 / log 2

log x / log 4 = log 5 / log 2 - log x / log 2

log x / log 4 = (log 5 - log x) /log 2

log x = (log 5 - log x) * log 4 / log 2

note that log 4 / log 2 = log (base2) 4 = 2

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

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

log x + log (x^2) = log 25

log [x(x^2)] = log 25

log (x^3) = log 25

x^3 = 25

x = 25^(1/3) = 2.924

log(base3)x=5log(base10)2

log(base3)x = 5 log 2 = log (2^5)

log x / log 3 = log 25

log x = log 25 * log 3

log x = 0.6670 = log (10^0.6670)

x = 10^0.6670 = 4.6450