Condense the logarithm

Question

(12 log x + 4/5 log y + 3 log x) - (log z + 2/5 log h + 8/5 log g)

Answer 1

like terms= 15logx+4/5logy-logz-2/5logh-8/5logg

Answer 2

Condensing logarithms are SO fun! (I know, I know, nerd alert!) The first thing to tackle is the numbers in front of the logs. When a number is in front of a log, it's actually going to be turned into an exponent when condensed:

(12 log x + 4/5 log y + 3 log x) - (log z + 2/5 log h + 8/5 log g)

(logx¹² + logy^4/5 + logx³) - (logz + logh^2/5 + logg^8/5)

Next, I'm going to condense inside of the parentheses. When you add logarithms, you can condense them down into one log by multiplication:

(logx¹² + logy^4/5 + logx³) - (logz + logh^2/5 + logg^8/5)

(logx¹²y^4/5x³) - (logzh^2/5g^8/5) **The first parentheses has two "x" variables so we need to combine them by adding the exponents

(logx¹⁵y^4/5) - (logzh^2/5g^8/5)

For our last step, we need to condense the two separate logs in the parentheses into one log. Since they are being subtracted, we are going to condense them down into division:

log(x¹⁵y^4/5/zh^2/5g^8/5)

And that's it! If you have additional questions don't hesitate to reach out! :)