1.9 Laws of logarithmsIB Maths: Applications and Interpretation HL: Revision notes

A logarithm answers the question 'what power of $a$ gives $x$?'. For $a>0$, $x>0$: $\log_a x=y\iff a^{y}=x.$ In examinations the base $a$ is $10$ or $e$. $\log_{10}x$ is often written $\lg x$, and $\log_e x=\ln x$ is the natural logarithm. Useful results follow straight from the definition: $\log_a1=0$, $\log_aa=1$, $\ln e^{x}=x$, $e^{\ln x}=x$ and $10^{\log_{10}x}=x$. You can only take the logarithm of a positive number.

For $a,x,y>0$: $\log_a(xy)=\log_ax+\log_ay$ $\log_a\left(\frac xy\right)=\log_ax-\log_ay$ $\log_a\left(x^{m}\right)=m\log_ax$ They hold for any base, so use them with $\log_{10}$ and $\ln$. They are the logarithm versions of the laws of exponents: multiplying powers adds indices, dividing subtracts them, and a power of a power multiplies them.

Use the laws to write several logarithms as one, or to break one logarithm into parts. Example: if $a=\ln3$ and $b=\ln5$ then $\ln15=\ln3+\ln5=a+b,\qquad\ln\frac95=2\ln3-\ln5=2a-b,\qquad\ln\sqrt{45}=\frac12\left(2a+b\right)=a+\frac b2.$ Single logarithm: $2\ln x-\ln y+\ln3=\ln\frac{3x^{2}}{y}$. Always apply the power law first, then combine with the product and quotient laws.

When the unknown is in the index, take logarithms of both sides, then use the power law to bring it down. Example: $3^{x}=50\Rightarrow x\ln3=\ln50\Rightarrow x=\frac{\ln50}{\ln3}=3.56$. With base $e$ the work is shorter. $800e^{0.35t}=1600$ gives $e^{0.35t}=2$, so $0.35t=\ln2$ and $t=1.98$. Isolate the exponential term first, then take $\ln$. A GDC can check your answer by solving the original equation directly.

Logarithms compress huge ranges of values. On a $\log_{10}$ scale, multiplying a quantity by $10$ adds $1$. The decibel scale $L=10\log_{10}\left(\frac{I}{I_0}\right)$ is an example: multiplying $I$ by $100$ adds $10\log_{10}100=20$ dB. Logarithms also straighten curves. If $y=kx^{n}$ then $\ln y=\ln k+n\ln x$, a straight line of $\ln y$ against $\ln x$ with gradient $n$ and intercept $\ln k$. If $y=ab^{x}$ then $\ln y=\ln a+x\ln b$, a straight line of $\ln y$ against $x$. A fitted line such as $\ln R=0.74\ln m+3.2$ gives $n=0.74$ and $k=e^{3.2}=24.5$, so $R=24.5m^{0.74}$.