Condensing Log Expressions

Question

Express as a single logarithm

In 5+ 2In x + 3 In (x² + 5)

Answer 1

As log properties go, logarithms that are added together will multiply within the parenthesis, logarithms that are subtracted will divide within the parenthesis, and constants multiplied to logarithms become exponents within the parentheses.

For instance,

log(a)+log(b) = log(ab)

log(a)-log(b) = log(a/b)

c*log(a)= log(a^c)

With these rules in mind, lets look at the problem at hand: In 5+ 2In x + 3 In (x² + 5)

Right off the bat, we can apply the third rule and change the constants in front of the logarithm expression to an exponent within the parentheses.

The expression becomes ln(5) + ln(x^2) + ln((x^2+5)^3)

Now, we can apply the first rule, the addition of logarithms becomes the multiplication of the quantities within the parentheses.

The expression becomes ln(5x^2*(x^2+5)^3)

This is the final expression, as there is only one natural log remaining.