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Amy Ferguson Moncure

IB · Unit 1 · 1C Logarithms

Simplify Logarithmic Expressions

Combine like log terms, or cancel matching powers of log x or ln x in a fraction. Split a sum before you cancel.

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Key Ideas Math Facts Easy Medium Hard

\(\log 10^k = k\)

\(10^{\log a} = a\)

\(\ln e^k = k\)

\(e^{\ln a} = a\)

What to Remember

Like terms

Treat \(\log x\) the way you treat a variable.

\(\log x + 3\log x = 4\log x\)

Cancel one power

The formula is \(\dfrac{( \log a )^n}{\log a} = (\log a)^{n-1}\) when \(\log a \neq 0\).

\(\dfrac{3(\log 5x)^2}{\log 5x} = 3\log 5x\)

Split a sum over a log

\(\dfrac{\ln x + (\ln x)^2}{\ln x} = 1 + \ln x\)

Divide each term by the denominator.

Then subtract

After you split, combine like terms.

\(1 + \ln x - \ln x = 1\)

Math Facts

When a question writes log with no base, that means log10.

ln means loge, the natural logarithm.

A different base is written as a subscript, for example log2 8. This is read as log base 2 of 8.

Level 1

Easy

Combine like log terms.

Example
Simplify. \(\log x + 3\log x\)

Like terms: add the coefficients.

\(1\log x + 3\log x = 4\log x\)

Answer: \(4\log x\)

1
Simplify. \(2\log x + 5\log x\)
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\(7\log x\)

Answer: \(7\log x\)

2
Simplify. \(8\ln x - 3\ln x\)
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\(5\ln x\)

Answer: \(5\ln x\)

3
Simplify. \(6\log x - \log x\)
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\(5\log x\)

Answer: \(5\log x\)

4
Simplify. \(4\ln x + \ln x\)
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\(5\ln x\)

Answer: \(5\ln x\)

5
Simplify. \(9\log x - 4\log x\)
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\(5\log x\)

Answer: \(5\log x\)

Level 2

Medium

Cancel one matching factor of log or ln from numerator and denominator.

Example
Simplify. \(\dfrac{3(\log 5x)^2}{\log 5x}\)

\(\dfrac{(\log 5x)^2}{\log 5x} = \log 5x\)

\(3\log 5x\)

Answer: \(3\log 5x\)

1
Simplify. \(\dfrac{4(\ln x)^2}{\ln x}\)
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\(4\ln x\)

Answer: \(4\ln x\)

2
Simplify. \(\dfrac{\log 7x}{(\log 7x)^2}\)
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\(\dfrac{1}{\log 7x}\)

Answer: \(\dfrac{1}{\log 7x}\)

3
Simplify. \(\dfrac{5(\log 2x)^3}{(\log 2x)^2}\)
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\(5\log 2x\)

Answer: \(5\log 2x\)

4
Simplify. \(\dfrac{(\ln 4x)^2}{\ln 4x}\)
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\(\ln 4x\)

Answer: \(\ln 4x\)

5
Simplify. \(\dfrac{6\log x}{2\log x}\)
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\(3\)

Answer: \(3\)

Level 3

Hard

Split every term, cancel powers, then combine. Extra practice on rational expressions in ln x or log x.

Example
Simplify. \(\dfrac{2\ln x + (\ln x)^2}{\ln x} - \ln x\)

\(\dfrac{2\ln x}{\ln x} + \dfrac{(\ln x)^2}{\ln x} - \ln x\)

\(2 + \ln x - \ln x\)

\(2\)

Answer: \(2\)

1
Simplify. \(\dfrac{3\log x + (\log x)^2}{\log x} - \log x\)
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\(3 + \log x - \log x\)

\(3\)

Answer: \(3\)

2
Simplify. \(\dfrac{\ln x - (\ln x)^3}{(\ln x)^2}\)
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\(\dfrac{\ln x}{(\ln x)^2} - \dfrac{(\ln x)^3}{(\ln x)^2}\)

\(\dfrac{1}{\ln x} - \ln x\)

Answer: \(\dfrac{1}{\ln x} - \ln x\)

3
Simplify. \(\dfrac{(\log x)^2 + 5\log x}{\log x}\)
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\(\log x + 5\)

Answer: \(\log x + 5\)

4
Simplify. \(\dfrac{4(\ln x)^3 - 2(\ln x)^2}{(\ln x)^2}\)
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\(4\ln x - 2\)

Answer: \(4\ln x - 2\)

5
Simplify. \(\dfrac{(\ln x)^3 + 2(\ln x)^2}{(\ln x)^2} - \ln x\)
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\(\ln x + 2 - \ln x\)

\(2\)

Answer: \(2\)

6
Simplify. \(\dfrac{6\log x - 2(\log x)^2}{\log x}\)
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\(6 - 2\log x\)

Answer: \(6 - 2\log x\)

7
Simplify. \(\dfrac{(\ln x)^2 - 4\ln x}{\ln x} + 4\)
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\(\ln x - 4 + 4\)

\(\ln x\)

Answer: \(\ln x\)

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