We start with log_5(2x+15)=log_5(3x). In this case we got lucky and got an equation that has the same base, 5. We can use this to our adantage, as long as we know how to convert a logarithm to an exponent. We do that using this rule: log_color(red)(b)color(blue)(x)=color(green)(y) becomes color(red)(b)^color(green)(y)=color(blue)(x). The base stays the same and x and y switch.
Let's set up our conversion: log_color(red)(5)color(blue)(2x+15)=color(green)(log_5(3x)) becomes color(red)(5)^color(green)(log_5(3x))=color(blue)(2x+15). Now that we've got this set up, let's clean it up. Notice the color(red)(5)^color(green)(log_5) part. These two are inverses of eachother, so they cance out. That leaves us with cancel(color(red)(5))^cancel(color(green)(log_5))color(white)(1)^color(green)(3x)=color(blue)(2x+15). The 3x dops down and gives us 3x=2x+15. That can be simplifed by subtracting 2x on both sides, which leaves us with x=15.
