# Thread: Proving Law of Exp and Log

1. ## Proving Law of Exp and Log

1. Given the product law of logarithms, prove the product law of exponents.

2. Given the quotient law of logarithms, prove the quotient law of exponents.

3. Apply algebraic reasoning to show that
a=b^(loga/logb) for any a,b>0

All I know is that

The product law of logs are:

Log(AB)=logA+logB

The Quotient law of logs are:

Log(A/B)=logA-Logb

Edit:

For product law:
let m=b^x and let n=b^y
mn=(b^x )(b^y)
and then what should i do?

2. Originally Posted by skeske1234
...

3. Apply algebraic reasoning to show that
a=b^(loga/logb) for any a,b>0

...
I assume that loga means $\log_{10}(a)$. If so:

You are supposed to know: $a = 10^{\log_{10}(a)}$

$a=b^{\dfrac{\log_{10}(a)}{\log_{10}(b)}}$

$a^{\log_{10}(b)}=\left( b^{\dfrac{\log_{10}(a)}{\log_{10}(b)}} \right)^{\log_{10}(b)}$

$a^{\log_{10}(b)}= b^{\log_{10}(a)}$

Now express the values of a and b as a power to the base 10:

$\left(10^{\log_{10}(a)} \right)^{\log_{10}(b)} = \left(10^{\log_{10}(b)} \right)^{\log_{10}(a)}~\implies~\boxed{10^{{\log_{ 10}(a)} \cdot {\log_{10}(b)}} = 10^{{\log_{10}(b)} \cdot {\log_{10}(a)}}}$