PLS HELP! qn on logarithms!change of base!?

Question

how to change base for this qn? solve it withought the use of calculators.


log5 4. log2 10/ log25 (square root 10)

ans: 2

Answer 1

If this is log base 5 of 4 times log base 2 of 10 all divided by log base 25 of the square root of 10... I'm getting that the answer is 8, not 2...

I tend to use natural logs (LN)...
and the change of base formula is log[B]X = LN X / LN B...

... log base 5 of 4 can become LN 4 / LN 5
LN 4 can be rewritten as LN 2^2... and then you can move down the 2... so it's really 2*LN2

... log base 2 of 10 can become LN 10 / LN 2

... log base 25 of the square root of 10 can become LN(sqrt10) / LN25. The LN 25 can be rewritten as LN 5^2 and then the 2 can come down in front so you really have 2 LN5. The LN (sqrt 10) can be rewritten as LN 10^(1/2) and the 1/2 can come down in front so you really have 1/2 LN10. *keep in mind that this part was the denominator so it's really flipped over when we change it to multiplication, the 1/2LN10 is on bottom and the 2LN5 is on top.

In the top we've got 2LN2 * LN10 * 2LN5
On the bottom we've got LN5 * LN2 * 1/2 LN10

The LN2 can cancel, the LN5 can cancel, the LN10 can cancel...

... so we have 2*2 / (1/2) = 4 / (1/2) = 4/1*2/1 = 8

Answer 2

log5 4. log2 10/ log25 (square root 10) =
2*log5 (4)* log2 (10)/ log25 (10) =
2(log(4) / log(5))(log(10) / log(2)) / (log(10) / log(25)) =
2(log(4) / log(5))(1 / log(2)) log(25) =
2(log(4) / log(2) )(log(25) / log(5) ) =
2(2log(2) / log(2) )(2log(5) / log(5) ) =
2(2)(2) =
8

not 2

Answer 3

use the change of base formula. It has the power to do many amazing...things.

Answer 4

8 is the answer