IB AA ยท 12B Logarithms
Convert log_b (expression) = k into exponential form, then solve. Check the argument.
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\(\log_b(b^k)=k\)
\(\log A+\log B=\log(AB)\)
\(\ln a^n=n\ln a\)
\(\log_b a=\dfrac{\ln a}{\ln b}\)
Convert to exponential form
The formula is \(\log_b x=k\) means \(x=b^k\).
\(\log_2 x=5\)
\(x=2^5\)
\(x=32\)
A linear argument
Convert first, then solve for x.
\(\log_2(x-3)=4\)
\(x-3=2^4\)
\(x-3=16\)
\(x=19\)
A fractional exponent
\(\log_9 x=\dfrac{1}{2}\) means \(x=9^{1/2}=3\).
Check the argument
The input of a log must be positive. After you solve, check that the expression inside the log is greater than 0.
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 \(\log_2 8\).
Convert log_b x = k into x = b^k.
\(\log_b x=k\) means \(x=b^k\)
\(x=2^5\)
\(x=32\)
Answer: \(32\)
\(x=5^3\)
\(x=125\)
Answer: \(125\)
\(x=2^0\)
\(x=1\)
Answer: \(1\)
\(x=4^1\)
\(x=4\)
Answer: \(4\)
\(x=10^{-1}\)
\(x=0.1\)
Answer: \(0.1\)
\(x=9^{1/2}\)
\(x=3\)
Answer: \(3\)
The argument is linear. Convert, then solve.
\(x-3=2^4\)
\(x-3=16\)
\(x=19\)
Answer: \(19\)
\(x+1=3^2\)
\(x+1=9\)
\(x=8\)
Answer: \(8\)
\(2x=5^1\)
\(2x=5\)
\(x=\dfrac{5}{2}\)
Answer: 52
\(x-2=4^0\)
\(x-2=1\)
\(x=3\)
Answer: \(3\)
\(3x+1=2^3\)
\(3x+1=8\)
\(3x=7\)
\(x=\dfrac{7}{3}\)
Answer: 73
\(x-4=3\)
\(x=7\)
Answer: \(7\)
A negative right-hand side. Check that the argument stays positive.
\(x-1=2^{-3}\)
\(x-1=\dfrac{1}{8}\)
\(x=\dfrac{9}{8}\)
Answer: 98
\(2x-5=3^{-1}\)
\(2x-5=\dfrac{1}{3}\)
\(2x=\dfrac{16}{3}\)
\(x=\dfrac{8}{3}\)
Answer: 83
\(x+6=4^{1/2}\)
\(x+6=2\)
\(x=-4\)
Answer: \(-4\)
\(5-x=2^{-1}\)
\(5-x=\dfrac{1}{2}\)
\(x=\dfrac{9}{2}\)
Answer: 92
\(3x+1=5^{-2}\)
\(3x+1=\dfrac{1}{25}\)
\(3x=-\dfrac{24}{25}\)
\(x=-\dfrac{8}{25}\)
Answer: \(-\dfrac{8}{25}\)
\(x-3=8^{2/3}\)
\(x-3=4\)
\(x=7\)
Answer: \(7\)
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