Math Notes for Problem Solving

Logarithms (Logs)

Formula:
Given base $b$, where $b>0$ and $b\ne 1$:

$$b^y=x$$

Then:

$${\log }_b(x)=y$$

Properties of logs:

1. Product Rule

$${\log }_b(x\cdot y)={\log }_b(x)+{\log }_b(y)$$

The logarithm of a product is the sum of the logarithms.

Example:

$${\log }_2(8)=3,{\log }_2(32)=5,8\cdot 3=256,{\log }_2(256)=8$$$${\log }_2(8)+{\log }_2(32)={\log }_2(256)$$

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2. Quotient Rule

$${\log }_b\left(\frac{x}{y}\right)={\log }_b(x)-{\log }_b(y)$$

The logarithm of a quotient is the difference of the logarithms.

Example:

$${\log }_2(16)=4,{\log }_2(4)=2,\frac{16}{4}=4,{\log }_2(4)=2$$$${\log }_2(16)-{\log }_2(4)={\log }_2(4)$$

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3. Power Rule

$${\log }_b(x^k)=k\cdot {\log }_b(x)$$

The logarithm of a number raised to an exponent is the exponent times the logarithm.

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4. Change of Base Formula

$${\log }_b(x)=\frac{{\log }_k(x)}{{\log }_k(b)}$$

You can change the base of a logarithm using any positive base $k\ne 1$.

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5. Logarithm of 1

$${\log }_b(1)=0$$

The logarithm of 1 to any base is 0.

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6. Logarithm of the Base

$${\log }_b(b)=1$$

The logarithm of a base to itself is 1.

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7. Inverse Property

$$b^{{\log }_b(x)}=x\text{and}{\log }_b(b^x)=x$$

The base raised to its logarithm cancels out, leaving the argument $x$.

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8. Equality Property

If:

$${\log }_b(x)={\log }_b(y)\$\$Then:\$\$x=y$$

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9. Logarithmic Identity

For natural logarithms and exponential functions:

$$e^{\ln (x)}=x\text{and}\ln (e^x)=x$$

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10. Base Conversion

You can express logarithms in terms of common logarithms (${\log }_{10}$) or natural logarithms ($\ln$):

$${\log }_b(x)=\frac{\ln (x)}{\ln (b)}\text{or}{\log }_b(x)=\frac{{\log }_{10}(x)}{{\log }_{10}(b)}$$

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11. Multiplicative Inverse Property

$${\log }_b\left(\frac{1}{x}\right)=-{\log }_b(x)$$

The logarithm of a reciprocal is the negative logarithm.

Applications on Logarithms

Finding Mulitplication by Logarithms

Formula:

$$x\cdot y=e^{\ln (x\cdot y)}$$$$\ln (x\cdot y)=\ln (x)+\ln (y)$$$$x\cdot y=e^{\ln (x)+\ln (y)}$$

Or:

1. Find ${\log }_b(x)$ and ${\log }_b(y)$, then add them.
2. Find the anit-log of thier sum. Raise the base to thier sum. $b^{{\log }_b(x)+{\log }_b(y)}$

Example: Find $5\times 6$ using logarithms:

Note: $\log ={\log }_{10}$

$$\log (5)\approx 0.6989,\log (6)\approx 0.7781$$$$\log (5)+\log (6)\approx 1.4771$$$${10}^{\log (5)+\log (6)}={10}^{1.4771}=30$$

Finding Division by Logarithms

Formula:

$$x÷y=e^{\ln \left(\frac{x}{y}\right)}$$$$\ln \left(\frac{x}{y}\right)=\ln (x)-\ln (y)$$$$x÷y=e^{\ln (x)-\ln (y)}$$

Or:

1. Find ${\log }_b(x)$ and ${\log }_b(y)$, then subtract them.
2. Find the anit-log of thier difference. Raise the base to thier subtraction. $b^{{\log }_b(x)\text{−}{\log }_b(y)}$

Example: Find $50÷5$ using logarithms:

$${\log }_2(50)\approx 5.6438,{\log }_2(5)\approx 2.3219$$$${\log }_2(50)\text{−}{\log }_2(5)\approx 3.3219$$$$2^{{\log }_2(50)\text{−}{\log }_2(5)}=2^{3.3219}=10$$