Express the Logarithm of a single expression

Question

Express the logarithm of a single expression. Assume that variables represent positive numbers.

Log₁₀²⁵+log₁₀³_log₁₀¹⁵

Answer 1

Here you are using the following 2 log rules to simplify the expression:

1) log_x(a) + log_x(b) = log_x(a•b)

2) log_x(a) – log_x(b) = log_x(a÷b)

So, from rule #1 (above), log₁₀(25) + log₁₀(3) = log₁₀(25•3) = log₁₀(75)

This leaves, log₁₀(75) – log₁₀(15), and from rule #2 (above), log₁₀(75) – log₁₀(15) = log₁₀(75÷15)

Simplify: log₁₀(75÷15) = log₁₀(5)

Hope this helps,

Todd

Answer 2

There are two basic rules applicable here:

1) log_b(m) + log_b(n) = log_b(m•n)

2) log_b(m) - log_b(n) = log_b(m/n)

If you group the first two elements of the problem together like this:

[log₁₀(25) + log₁₀(3)] - log₁₀(15)

and use rule #1 you get:

log₁₀(25•3) - log₁₀(15)

which simplifies to:

log₁₀(75) - log₁₀(15)

then apply rule #2 and you get:

log₁₀(75/15)

and then simplify, and you get:

log₁₀(5)