I have had a problem with logarithm problems and one in particular; the problem states: log 5^x+1= log 3^x. Can you please help me with this problem and explain the steps as well? Thank you.
$\displaystyle \log 5^x+1=\log 3^x$, by the exponent rule for logarithms you may "bring down" the $\displaystyle x$ thus,Originally Posted by aussiekid90
$\displaystyle x\log 5+1=x\log 3$
Then,
$\displaystyle x\log 5-x\log 3=-1$
Thus,
$\displaystyle x(\log 5-\log 3)=-1$
Thus,
$\displaystyle x=-\frac{1}{\log 5-\log 3}=\frac{1}{\log 3-\log 5}$
Q.E.D.
Take the logarithm of both sides,Originally Posted by aussiekid90
$\displaystyle \log 5^{x+1}=\log 3^x$
Now use the exponent rule and "bring down" exponents,
$\displaystyle (x+1)\log 5=x\log 3$
Open parantheses,
$\displaystyle x\log 5+\log 5=x\log 3$
Rearrange,
$\displaystyle x\log 5-x\log 3=-\log 5$
Factor,
$\displaystyle x(\log 5-\log 3)=-\log 5$
Thus,
$\displaystyle x=-\frac{\log 5}{\log 5-\log 3}\approx -3.15$
Q.E.D.