Exponents, Exponentials and Logarithms · Lesson 4

Log properties

Three rules, each a copy of an exponent rule. For positive MM and NN: the product rule log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N, the quotient rule log⁡bMN=log⁡bM−log⁡bN\log_b \frac{M}{N} = \log_b M - \log_b N, and the power rule log⁡bMk=klog⁡bM\log_b M^k = k\log_b M. Adding exponents when you multiply powers is the same fact as the product rule.

Expand and condense. To expand, split products and quotients, then bring exponents to the front: log⁡525x3y=2+3log⁡5x−log⁡5y\log_5 \frac{25x^3}{y} = 2 + 3\log_5 x - \log_5 y. To condense, go the other way: move each coefficient up as an exponent first, then turn sums into products and differences into quotients.

Use given values by factoring. If you know log⁡b2\log_b 2 and log⁡b3\log_b 3, you can find log⁡b18\log_b 18: since 18=2⋅3⋅318 = 2 \cdot 3 \cdot 3, log⁡b18=log⁡b2+2log⁡b3\log_b 18 = \log_b 2 + 2\log_b 3.

Change of base. log⁡ba=log⁡alog⁡b\log_b a = \frac{\log a}{\log b} turns any log into common logs your calculator can do. Know it in case a question needs it without stating it.

Worked example Easy

For x>0x > 0, which expression is equivalent to log⁡2(8x5)\log_2(8x^5)?

  1. 3+5log⁡2x3 + 5\log_2 xAnswer
  2. 8+5log⁡2x8 + 5\log_2 x
  3. 15log⁡2x15\log_2 x
  4. 5log⁡2(8x)5\log_2(8x)

How to solve it

  1. Product rule: log⁡2(8x5)=log⁡28+log⁡2x5\log_2(8x^5) = \log_2 8 + \log_2 x^5.
  2. log⁡28=3\log_2 8 = 3, because 23=82^3 = 8.
  3. Power rule: log⁡2x5=5log⁡2x\log_2 x^5 = 5\log_2 x.
  4. Together: 3+5log⁡2x3 + 5\log_2 x.

Why each choice is right or wrong

  • A. Correct. log⁡2(8x5)=log⁡28+log⁡2x5=3+5log⁡2x\log_2(8x^5) = \log_2 8 + \log_2 x^5 = 3 + 5\log_2 x.
  • B. Incorrect. 8+5log⁡2x8 + 5\log_2 x leaves 8 where log⁡28\log_2 8 belongs. The product rule gives log⁡28\log_2 8, which equals 3.
  • C. Incorrect. 15log⁡2x15\log_2 x multiplies log⁡28=3\log_2 8 = 3 by 5. The product rule adds the logs of the factors; it doesn't multiply them.
  • D. Incorrect. 5log⁡2(8x)5\log_2(8x) applies the exponent 5 to the 8 as well. Only xx is raised to the 5th power, so 8x58x^5 is not (8x)5(8x)^5.

Worked example Medium

For positive xx, yy and zz, which expression is equivalent to 2log⁡x+3log⁡y−log⁡z2\log x + 3\log y - \log z?

  1. log⁡x2y3z\log \frac{x^2y^3}{z}Answer
  2. log⁡6xyz\log \frac{6xy}{z}
  3. log⁡(x2+y3−z)\log(x^2 + y^3 - z)
  4. log⁡(x2y3)log⁡z\frac{\log (x^2y^3)}{\log z}

How to solve it

  1. Power rule first: 2log⁡x=log⁡x22\log x = \log x^2 and 3log⁡y=log⁡y33\log y = \log y^3.
  2. Product rule: log⁡x2+log⁡y3=log⁡(x2y3)\log x^2 + \log y^3 = \log(x^2y^3).
  3. Quotient rule: subtracting log⁡z\log z divides by zz, giving log⁡x2y3z\log \frac{x^2y^3}{z}.

Why each choice is right or wrong

  • A. Correct. The coefficients become exponents, the sum becomes a product and the difference becomes a quotient: log⁡x2y3z\log \frac{x^2y^3}{z}.
  • B. Incorrect. log⁡6xyz\log \frac{6xy}{z} multiplies the coefficients 2 and 3 into the argument. A coefficient in front of a log becomes an exponent: 2log⁡x=log⁡x22\log x = \log x^2.
  • C. Incorrect. log⁡(x2+y3−z)\log(x^2 + y^3 - z) treats a sum of logs as the log of a sum. Adding logs multiplies the inputs, and subtracting a log divides.
  • D. Incorrect. log⁡(x2y3)log⁡z\frac{\log (x^2y^3)}{\log z} turns the subtraction into a division of logs. log⁡M−log⁡N=log⁡MN\log M - \log N = \log \frac{M}{N}, which is not log⁡Mlog⁡N\frac{\log M}{\log N}.

Trap There is no rule for the log of a sum

log⁡(a+b)\log(a + b) can't be split. log⁡2+log⁡5=log⁡10=1\log 2 + \log 5 = \log 10 = 1, but log⁡(2+5)=log⁡7\log(2 + 5) = \log 7, which is about 0.85. Only a product inside the log becomes a sum outside.

Trap A quotient of logs is not a difference

log⁡alog⁡b\frac{\log a}{\log b} is the change-of-base form of log⁡ba\log_b a. It is not log⁡(a−b)\log(a - b), and it is not log⁡ab\log \frac{a}{b} either; only log⁡a−log⁡b\log a - \log b equals log⁡ab\log \frac{a}{b}.

Calculator When a calculator isn’t faster

These questions are mostly about recognizing a rule, and hand work is quicker. To test an equivalence, graph the original and a choice in the built-in Desmos calculator if you test online, or evaluate both at a value such as x=2x = 2 on your calculator; equivalent expressions give the same number.

Pick a test value that keeps every log input positive, and use your calculator's log key for base 10.