IB AA ยท 12B Logarithms
When the bases differ, take ln of both sides. Collect x, then divide.
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\(\log_b(b^k)=k\)
\(\log A+\log B=\log(AB)\)
\(\ln a^n=n\ln a\)
\(\log_b a=\dfrac{\ln a}{\ln b}\)
Different bases
Take ln of both sides. Use \(\ln(a^n)=n\ln a\).
Collect x terms
\(5^x=2^{x+1}\)
\(x\ln 5=(x+1)\ln 2\)
Expand, then factor x.
Exact form
Leave the answer as a stacked fraction of ln values. Do not use a decimal unless the question asks for one.
Matching bases first
If both sides are powers of the same prime, rewrite first. \(8^{x}=4^{x+1}\) becomes \((2^3)^x=(2^2)^{x+1}\).
When a question writes \(\log\) with no base, that means \(\log_{10}\). \(\ln\) means \(\log_e\), the natural logarithm.
A different base is written as a subscript, for example \(\log_2 8\).
Take ln of both sides. Leave x in terms of ln.
\(\ln(a^n)=n\ln a\)
\(x\ln 5=(x+1)\ln 2\)
\(x\ln 5=x\ln 2+\ln 2\)
\(x\ln 5-x\ln 2=\ln 2\)
\(x(\ln 5-\ln 2)=\ln 2\)
\(x=\dfrac{\ln 2}{\ln 5-\ln 2}\)
Answer: \(\dfrac{\ln 2}{\ln 5-\ln 2}\)
\(x\ln 3=(x-1)\ln 7\)
\(x\ln 3=x\ln 7-\ln 7\)
\(x\ln 7-x\ln 3=\ln 7\)
\(x=\dfrac{\ln 7}{\ln 7-\ln 3}\)
Answer: \(\dfrac{\ln 7}{\ln 7-\ln 3}\)
\(x\ln 4=(x+2)\ln 3\)
\(x\ln 4-x\ln 3=2\ln 3\)
\(x=\dfrac{2\ln 3}{\ln 4-\ln 3}\)
Answer: \(\dfrac{2\ln 3}{\ln 4-\ln 3}\)
\((x+3)\ln 2=x\ln 5\)
\(x\ln 2+3\ln 2=x\ln 5\)
\(3\ln 2=x\ln 5-x\ln 2\)
\(x=\dfrac{3\ln 2}{\ln 5-\ln 2}\)
Answer: \(\dfrac{3\ln 2}{\ln 5-\ln 2}\)
\((x-2)\ln 6=x\ln 2\)
\(x\ln 6-2\ln 6=x\ln 2\)
\(x\ln 6-x\ln 2=2\ln 6\)
\(x=\dfrac{2\ln 6}{\ln 6-\ln 2}\)
Answer: \(\dfrac{2\ln 6}{\ln 6-\ln 2}\)
\((x+1)\ln 3=(x-1)\ln 4\)
\(x\ln 3+\ln 3=x\ln 4-\ln 4\)
\(x\ln 4-x\ln 3=\ln 3+\ln 4\)
\(x=\dfrac{\ln 3+\ln 4}{\ln 4-\ln 3}\)
Answer: \(\dfrac{\ln 3+\ln 4}{\ln 4-\ln 3}\)
Rewrite both sides as powers of 2 or 3 first when you can.
\(8=2^3\)
\(4=2^2\)
\((2^3)^x=(2^2)^{x+1}\)
\(2^{3x}=2^{2x+2}\)
\(3x=2x+2\)
\(x=2\)
Answer: \(2\)
\(9=3^2\)
\(27=3^3\)
\(3^{2x}=3^{3x-3}\)
\(2x=3x-3\)
\(x=3\)
Answer: \(3\)
\(\sqrt{2}=2^{1/2}\)
\(8=2^3\)
\(2^{x/2}=2^{3x-3}\)
\(\dfrac{x}{2}=3x-3\)
\(x=6x-6\)
\(x=\dfrac{6}{5}\)
Answer: 65
\(\sqrt{3}=3^{1/2}\)
\(9=3^2\)
\(3^{x/2}=3^{2x-4}\)
\(\dfrac{x}{2}=2x-4\)
\(x=4x-8\)
\(x=\dfrac{8}{3}\)
Answer: 83
\(2^{2x+2}=2^{3x-3}\)
\(2x+2=3x-3\)
\(x=5\)
Answer: \(5\)
\(2^{4x}=2^{x+6}\)
\(4x=x+6\)
\(x=2\)
Answer: \(2\)
Different bases and a linear exponent on both sides. Leave x in terms of ln.
\(2x\ln 7=(x+4)\ln 3\)
\(2x\ln 7=x\ln 3+4\ln 3\)
\(2x\ln 7-x\ln 3=4\ln 3\)
\(x(2\ln 7-\ln 3)=4\ln 3\)
\(x=\dfrac{4\ln 3}{2\ln 7-\ln 3}\)
Answer: \(\dfrac{4\ln 3}{2\ln 7-\ln 3}\)
\((x-1)\ln 3=(x+2)\ln 5\)
\(x\ln 3-\ln 3=x\ln 5+2\ln 5\)
\(x\ln 5-x\ln 3=-\ln 3-2\ln 5\)
\(x=\dfrac{\ln 3+2\ln 5}{\ln 3-\ln 5}\)
Answer: \(\dfrac{\ln 3+2\ln 5}{\ln 3-\ln 5}\)
\((3x-1)\ln 2=\ln 11\)
\(3x-1=\dfrac{\ln 11}{\ln 2}\)
\(x=\dfrac{1}{3}\left(\dfrac{\ln 11}{\ln 2}+1\right)\)
Answer: \(\dfrac{1}{3}\left(\dfrac{\ln 11}{\ln 2}+1\right)\)
\((5-3x)\ln 2=(2x-1)\ln 3\)
\(5\ln 2-3x\ln 2=2x\ln 3-\ln 3\)
\(5\ln 2+\ln 3=2x\ln 3+3x\ln 2\)
\(x=\dfrac{5\ln 2+\ln 3}{3\ln 2+2\ln 3}\)
Answer: \(\dfrac{\ln(32\cdot 3)}{\ln(8\cdot 9)}\)
\((2x+3)\ln 5=(x-1)\ln 9\)
\(2x\ln 5+3\ln 5=x\ln 9-\ln 9\)
\(2x\ln 5-x\ln 9=-3\ln 5-\ln 9\)
\(x=\dfrac{-3\ln 5-\ln 9}{2\ln 5-\ln 9}\)
Answer: \(\dfrac{3\ln 5+\ln 9}{\ln 9-2\ln 5}\)
\((x+2)\ln 4=2x\ln 7\)
\(x\ln 4+2\ln 4=2x\ln 7\)
\(2\ln 4=2x\ln 7-x\ln 4\)
\(x=\dfrac{2\ln 4}{2\ln 7-\ln 4}\)
Answer: \(\dfrac{2\ln 4}{2\ln 7-\ln 4}\)
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