We need to solve the next expression using the properties of the logarithm:
[tex]\log _56\cdot\log _625[/tex]
Use the next logarithm property on both:
[tex]\log _ab\text{ =}\frac{\ln b}{\ln a}[/tex]
So:
[tex]\frac{\ln6}{\ln\text{ 5}}\cdot\frac{\ln 25}{\ln 6}[/tex]
Cancel the like terms, in this case, ln 6
Then:
[tex]\frac{\ln \text{ 25}}{\ln 5}[/tex]
Rewrite the expression ln 25, using the next property:
[tex]\ln x^{b\text{ }}=\text{ b}\cdot\ln \text{ x}[/tex]
Then
[tex]\ln 25\text{ = }\ln 5^2=2\ln 5[/tex]
Simplify the like terms:
[tex]\frac{2\ln 5}{\ln 5}=2[/tex]
Therefore, The result is 2.
