Question

How do you solve `log_(6)x + log_(6)3 = 2`?

Answer 1

`x=12`

Explanation:

We can start by subtracting `log_6(3)` from both sides. This gives us

`log_6x=color(darkviolet)(2-log_6(3))`

We can apply the logarithm rule

`a=log_b(b^a)`

where our `a` is what I have in purple above. This can be rewritten as

`log_6x=color(darkviolet)(log_6(ul(6^(2-log_6(3))))`

Let's simplify what I have underlined:

`6^(2-log_6(3))` can be rewritten as

`6^(-log_6(3))*6^2`

Which simplifies to

`3^-1*color(blue)(6^2)`

Further simplifying, we get

`1/3*color(blue)(36)=36/3=color(darkviolet)(12)`

Remember, this is the underlined portion of our equation. We have

`log_6x=color(darkviolet)(log_6(12))`

Since the bases are the same, we now have

`x=12`

All we did was:

• Subtract to get logs on both sides
• Rewrite the expression in a form we can deal with
• Simplify the expression using many exponent properties
• Realized that the bases are the same, so they basically cancel

Hope this helps!