IB AA ยท 12B Logarithms
Take log of both sides. Give x to 3 significant figures.
Jump to section:
\(\log_b(b^k)=k\)
\(\log A+\log B=\log(AB)\)
\(\ln a^n=n\ln a\)
\(\log_b a=\dfrac{\ln a}{\ln b}\)
Take log of both sides
\(a^x=n\) becomes \(x\log a=\log n\), so \(x=\dfrac{\log n}{\log a}\).
A linear exponent
\(3^{x+2}=14\)
\(x+2=\dfrac{\log 14}{\log 3}\)
Then subtract 2.
Three significant figures
Round the calculator display to 3 significant figures unless the question asks for an exact answer.
Any base of log
LOG or LN both work. Stay with one of them for the whole calculation.
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 \(\log_2 8\).
Exact answers stay in log or ln form. Use technology only when the question asks for a decimal.
Take log of both sides. Give x to 3 significant figures.
\(x\log 2=\log 13\)
\(x=\dfrac{\log 13}{\log 2}\)
\(x=3.70\)
Answer: \(3.70\)
\(x=\dfrac{\log 20}{\log 5}\)
\(x=1.86\)
Answer: \(1.86\)
\(x=\dfrac{\log 11}{\log 6}\)
\(x=1.34\)
Answer: \(1.34\)
\(x=\dfrac{\log 15}{\log 4}\)
\(x=1.95\)
Answer: \(1.95\)
\(x=\log 7\)
\(x=0.845\)
Answer: \(0.845\)
\(x=\dfrac{\log 40}{\log 3}\)
\(x=3.36\)
Answer: \(3.36\)
The exponent is linear. Isolate the power, then take logs.
\(x+2=\dfrac{\log 14}{\log 3}\)
\(x+2=2.40\)
\(x=0.402\)
Answer: \(0.402\)
\(x-1=\dfrac{\log 18}{\log 5}\)
\(x-1=1.80\)
\(x=2.80\)
Answer: \(2.80\)
\(x+3=\dfrac{\log 25}{\log 2}\)
\(x+3=4.64\)
\(x=1.64\)
Answer: \(1.64\)
\(x+1=\dfrac{\log 40}{\log 7}\)
\(x+1=1.90\)
\(x=0.896\)
Answer: \(0.896\)
\(x-2=\dfrac{\log 9}{\log 4}\)
\(x-2=1.58\)
\(x=3.58\)
Answer: \(3.58\)
\(x-2=\dfrac{\log 8}{\log 3}\)
\(x-2=1.89\)
\(x=3.89\)
Answer: \(3.89\)
A longer linear exponent. Still isolate, then take logs.
\(2x-1=\dfrac{\log 9}{\log 4}\)
\(2x-1=1.58\)
\(2x=2.58\)
\(x=1.29\)
Answer: \(1.29\)
\(2x+1=\dfrac{\log 20}{\log 3}\)
\(2x+1=2.73\)
\(x=0.863\)
Answer: \(0.863\)
\(3x-2=\dfrac{\log 19}{\log 2}\)
\(3x-2=4.25\)
\(x=2.08\)
Answer: \(2.08\)
\(x-2=\dfrac{\log 8}{\log 5}\)
\(x-2=1.29\)
\(x=3.29\)
Answer: \(3.29\)
\(4x+1=\dfrac{\log 30}{\log 2}\)
\(4x+1=4.91\)
\(x=0.977\)
Answer: \(0.977\)
\(2x-3=\dfrac{\log 15}{\log 6}\)
\(2x-3=1.51\)
\(x=2.25\)
Answer: \(2.25\)
Need help solving exponential equations with a calculator?
Book a Session with Amy