MHA2: Properties of Logarithms

Now, we'll proceed to the properties of logarithms. If you need assistance with taking the logarithm of a number to a certain base, please feel free to check MHA1. Okay, so what are the properties of logs? Here are some of them.

#1 log_baˣ = xlog_ba
#2 log_b(ax) = log_ba + log_bx
#3 log_b(a/x) = log_ba - log_bx
#4 log_b1 = 0
#5 log_bx = logx / logb

Also, since we're dealing with real numbers, remember these two:
1. You can't take the logarithm of 0.
2. You can't take the logarithm of a negative number.

Familiarize yourself with these things, because these are your weapons in solving logarithmic equations. Let's have some examples where these properties are used.

Example 1: Express -log m + 7 log(6 + n) as a single logarithm.
= -log m + 7log(6 + n)

= 7log(6 + n) - logm
= log(6 + n)⁷ - logm (Property #1)
= log[(6 + n)⁷ / m] (Property #3)

Example 2: Simplify logx + log(3x) - log(5x²)
= logx + log(3x) - log(5x²)
= log(x * 3x) - log(5x²) (Property #2)
= log(3x²) - log(5x²) 
= log(3x² / 5x²) (Property #3)
= log(3/5) (Answer)

Example 3: Express log₂₇m - log₉m as a single logarithm.
= log₂₇m - log₉m
= (logm / log27) - (logm / log9) (Property #5)
= (logm / log3³) - (logm / log3²) 
= (logm / 3log3) - (logm / 2log3) (Property #1)
= (2logm / 6log3) - (3logm / 6log3) 
= -logm / 6log3
= log(m⁻¹) / 6log3 (Property #1)
= (1/6) * log(1/m) / log3
= (1/6)log₃(1/m) (Property #5)
Our aim here is to just express log₂₇m - log₉m as a single logarithm, so my answer can't be the only answer. Try expressing my answer in different forms!

Example 4: Express 4logx + (1/3)log(√x) - 2log(x^3) as a single logarithm.
= 4logx + (1/3)log(√x) - 2log(x³)
= log(x⁴) + log([x^(½)]^⅓) - log([x³]²) (Property #1)
= log(x⁴) + log(x^(⅙)) - log(x⁶)
= log(x⁴ * x^(⅙)) - log(x⁶) (Property #2)
= log(x^(25/6)) - log(x⁶)
= log(x^(25/6) / x⁶) (Property #3)
= log(1/x^(11/6)) 
= log(x^(-11/6))
= (-11/6)logx (Property #1)

I hope these examples have helped you. On MHA3, we'll now fully weaponize these properties of logarithms to solve logarithmic equations. Please stay tuned. By the way, just to keep you fired up, here's a problem.


Write the expression as sums and/or differences of logarithms without exponents.
log (m⁴ * ³√(n/p²)) The answer is 4logM + (1/3)logn - (2/3)logp