Compute the value of each expression without using a calculator.

1. $\log\left(8\right) + \log\left(125\right)$ 

   Step 1(a):

   Use the Product Property of Logs: $\log\left(N\right) + \log\left(M\right) = \log\left(NM\right)$.

   $\log(8 · 125)$

   Step 2(a):

   Remember you are working in log base $10$.

   $\log_{10}(1000)$

   Answer (a):

   $3$

2. $\log_{25}\left(125\right)$ 

   Step 1(b):

   Let the expression equal $x$ then convert to exponential form.

   $25^{x} = 125$

   Step 2(b):

   Rewrite each side of the equation as a power of $5$. The exponents are equal.

   $5^{2x} = 5^{3}$

3. $\frac { 1 } { 2 }\log(25) + \log(20)$ 

   Step 1(c):

   Apply the Power Property of Logs to the first logarithm and simplify.

   $\log (25)^{1/2} + \log (20)$

   $\log\left(5\right) + \log\left(20\right)$

   Step 2(c):

   Use the Product Property of Logs, as in part (a).

4. $7^{\log_7\left(12\right)}$ 

   Step (d):

   Remember the property that

   $b^{\text{log}_b(N)} = N$