log7^n=2/3log7^8 Please solve step by step, thanks

Question

campbell_st · Answer

ok... the left hand side can be rewritten using the index law for powers

$$\log(x^a) = a \times \log(x)$$

so you get 

$$n \times \log(7) = \frac{2}{3} \times 8\times  \log(7)$$

divide both sides of the equation log log(7)

and you'll have your answer

anonymous · Answer

n=4 ??

campbell_st · Answer

not quite,,, won't dividing by 7 leave

$$n = \frac{2}{3} \times 8$$

simplify the fraction

campbell_st · Answer

oops should read dividing by log(7)

anonymous · Answer

oh my friend told me it was 16/3 but i thought that was wrong.

campbell_st · Answer

nope its the correct answer

anonymous · Answer

Oh well i solved it a different way, I thought it would be log7^8^2/3=log7^4 which equals log7^n=log7^4 which makes n = 4 ? is that method wrong ?

campbell_st · Answer

well what you really have is 

$$\frac{2}{3} \log 7^8 = \log(7^8)^{\frac{2}{3}}$$

using the power of a power index law you get

$$\log 7 ^{8 \times \frac{2}{3}}$$


so n = 16/3

hope that makes sense...

anonymous · Answer

Ohhh so i had to multiply it and NOT raise by that power ?

campbell_st · Answer

well it uses a combination of index laws and log laws... 

so there are a few methods to solve the equation.

anonymous · Answer

Ohh ok, thanks so much for your help, I just have 1 more to do can i do it on my own and come back to you and tell me if the answer is right ?

campbell_st · Answer

just post it if you get stuck... and there are lots of people here who can help

anonymous · Answer

(2log6^4)-(1/4log6^16)=log6^x
x=2 can u just tell me if this is right its my last poblem.

anonymous · Answer

btw the problem didn;t include parenthesis i thought it would just help read the problem better.

anonymous · Answer

2log6^4-1/4log6^16=log6^x is the problem

campbell_st · Answer

ok.. so the log law for subtraction is 

$$\log(a) - \log(b) = \log(\frac{a}{b})$$

and you can rewrite the the powers

$$\log(\frac{(6^4)^2}{(6^{16})^{\frac{1}{4}}})$$

campbell_st · Answer

then simplify

anonymous · Answer

yeah i said its x =2 was just wanting to see if i was right lol

campbell_st · Answer

I would have said x = 4

$$\log \frac{6^8}{6^4} = \log(6^4)...then... \log(6^4) = \log(6^x)$$

anonymous · Answer

Ohhh ops forgot you subtract the power by the number being divided i just divided those 2, thanks for your help goodnight!!