--------------------------------------------------------------------------------

I have 4 logarithmic problems that I need help with...



log10 (10^1/2)



log10 (1/10^x)



log 3 x^2=2log3 4- 4log3 5



2 log4 9- log2 3

2 log_{4}(9)- log_{2}(3)

Try using the change of base formula. log_{a}b=\frac{log(b)}{log(a)}

2\frac{log(9)}{log(4)}-\frac{log(3)}{log(2)}

Property of logs:

\frac{4log(3)}{2log(2)}-\frac{log(3)}{log(2)}

Now, see it?

log_{10} 10^{\frac12} = \frac12 log_{10}10

From property of logs. Can you simplify now?

log_{10} \frac{1}{10^x}

Use the property of logs; log\frac ab = loga - logb

to give:

log_{10} \frac{1}{10^x} = log_{10}1 - log_{10}10^x

Then, simplify to log_{10}1 - log_{10}10^x = log_{10}1 - xlog_{10}10

Can you see it now?

log_3 x^2=2log_3 4- 4log_3 5

Use law of subtraction of logarithms, which I told you earlier. You'll have firstly:

log_3 x^2=log_3 4^2- log_3 5^4

then;

log_3 x^2=log_3 \frac{4^2}{5^4}

The logarithms cancel each other, leaving behind:

x^2=\frac{4^2}{5^4}

Can you solve it now?