# Thread: Logarithm Simplification Help. Any aid appreciated.

1. ## Logarithm Simplification Help. Any aid appreciated.

Hi everyone,

I have been working with introductory logarithms recently, and I came across these two problems that I just could not solve. They are:

The questions are:

Simplify

Log3 (4 to the power of 2) + 3 log3 2

4 log3 2 – 2 log3 6 + 2

Any help simplifying these equations will be very much appreciated!

Thanks,

Nathaniel

2. $\log_3{(4^2)} + 3\log_3{2} = \log_3{(2^4)} + 3\log_3{2}$

$= 4\log_3{2} + 3\log_3{2}$

$= 7\log_3{2}$.

3. Hello, BinaryBoy!

$\text{Simplify: }\;4\log_3(2) - 2\log_3(6) + 2$

$4\log_3(2) - 2\log_3(6) + 2 \;=\;\log_3\left(2^4\right) - \log_3\left(6^2\right) + 2$

. . . . . . . . . . . . . . . . . . $=\;\log_3(16)-\log_3(36) + 2$

. . . . . . . . . . . . . . . . . . $=\;\log_3\left(\frac{16}{36}\right) + 2$

. . . . . . . . . . . . . . . . . . $=\;\log_3\left(\frac{4}{9}\right) + 2$

. . . . . . . . . . . . . . . . . . $=\; \log_3(4) - \underbrace{\log_3(9)}_{\text{This is 2}} + 2$

. . . . . . . . . . . . . . . . . . $=\;\log_3(4) - 2 + 2$

. . . . . . . . . . . . . . . . . . $=\;\log_3(4)$

4. Thank you so much Prove It, and Soroban! That was a huge help!

But, just to clear up one point...Soroban, could you please clarify why log3 (9) becomes 2? Is it because of the "logarithm of the base" rule?

Cheers,

5. Originally Posted by BinaryBoy
... please clarify why log3 (9) becomes 2?
let $x = \log_3{9}$

then $3^x = 9$

$x = 2$

or ...

$\log_3{9} = \log_3(3^2) = 2\log_3{3} = 2 \cdot 1 = 2$

6. Thanks for clarifying, skeeter.