1) Laws of Exponents
Product
\( a^m \cdot a^n = a^{m+n} \)
Quotient
\( \dfrac{a^m}{a^n} = a^{m-n} \quad (a \ne 0) \)
Power of a Power
\( (a^m)^n = a^{mn} \)
Power of a Product
\( (ab)^n = a^n\, b^n \)
Power of a Quotient
\( \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \quad (b \ne 0) \)
Zero Exponent
\( a^0 = 1 \quad (a \ne 0) \)
Negative Exponent
\( a^{-n} = \dfrac{1}{a^n} \quad (a \ne 0) \)
Fractional Exponent
\( a^{\tfrac{m}{n}} = \sqrt[n]{a^m} = \big(\sqrt[n]{a}\big)^m \)
Tip: Same base ⇒ multiply/divide by adding/subtracting exponents. Different bases ⇒ try factoring or logs.
Worked Examples (Exponents)
Example A. Simplify \(2^3\cdot2^5\).
\(2^{3+5}=2^8=256\).
Example B. Simplify \(\dfrac{x^7}{x^2}\).
\(x^{7-2}=x^5\).
Example C. Simplify \((3a^2)^3\).
\(3^3 a^{2\cdot3}=27a^6\).
Example D. Write with positive exponents: \(y^{-4}\).
\(1/y^4\).
Example E. Evaluate \(16^{3/4}\).
\(16^{1/4}=2\Rightarrow 2^3=8\). So \(16^{3/4}=8\).
Common pitfalls (Exponents):
- \(a^m+a^n \neq a^{m+n}\) (no rule for addition).
- \((a+b)^n \neq a^n+b^n\) in general.
- \(a^0=1\) only for \(a\neq0\).
- Even roots need non-negative radicand to stay in the reals.
2) Laws of Logarithms
Definition: For base \(b>0,\; b\neq1\), \(y=\log_b(x)\) means \(b^y=x\). Logs and exponentials are inverses.
| Law | Statement | Idea |
|---|---|---|
| Product | \(\log_b(MN)=\log_b M+\log_b N\) | Multiply → add |
| Quotient | \(\log_b(M/N)=\log_b M-\log_b N\) | Divide → subtract |
| Power | \(\log_b(M^k)=k\,\log_b M\) | Exponent becomes factor |
| Identity | \(\log_b b=1,\; \log_b 1=0\) | Because \(b^1=b,\; b^0=1\) |
| Inverse | \(b^{\log_b x}=x,\; \log_b(b^x)=x\) | Undo each other |
Domain: \(\log_b x\) requires \(x>0\). Valid bases: \(b>0\) and \(b\neq 1\).
Change of Base
Formula
\(\displaystyle \log_b x=\frac{\log_k x}{\log_k b}\) for any base \(k\) (often \(10\) or \(e\)).
Worked Examples (Logs)
Example 1. Expand \(\log_3(27x^2)\).
\(\log_3 27+\log_3 x^2=3+2\log_3 x\) since \(3^3=27\).
Example 2. Condense \(2\log(5)-\log(2)\) (base 10).
\(\log(5^2)-\log(2)=\log\!\left(\dfrac{25}{2}\right)\).
Example 3. Solve \(3^{\,2x-1}=27\).
\(27=3^3\Rightarrow 2x-1=3\Rightarrow x=2\).
Example 4. Solve \(\log_2(5x)=3\).
\(2^3=5x\Rightarrow x=\dfrac{8}{5}=1.6\) (needs \(x>0\)).
Example 5. Evaluate \(\log_3(10)\) (calculator).
\(\log_3 10=\dfrac{\ln 10}{\ln 3}=\dfrac{\log 10}{\log 3}\).
Quick Practice
Exponents
- Simplify \(5^2\cdot5^4\).
- Simplify \((2x^3y)^2\).
- Write with positive exponents: \(a^{-3}b^{-2}\).
- Evaluate \(81^{3/4}\).
- \(5^6=15625\)
- \(4x^6y^2\)
- \(\dfrac{1}{a^3b^2}\)
- \((81^{1/4})^3=3^3=27\)
Logarithms
- Expand \(\ln(6x^2)\).
- Condense \(3\log_2 x-\log_2 4\).
- Solve \(10^x=7\).
- Solve \(\log_5(x-1)=2\).
- \(\ln 6+2\ln x\)
- \(\log_2(x^3)-\log_2 4=\log_2\!\left(\dfrac{x^3}{4}\right)\)
- \(x=\log_{10}7=\dfrac{\ln 7}{\ln 10}\)
- \(x-1=25\Rightarrow x=26\) (needs \(x>1\))
Summary (At a Glance)
Exponents
- \(a^m a^n=a^{m+n}\), \(\dfrac{a^m}{a^n}=a^{m-n}\)
- \((a^m)^n=a^{mn}\), \((ab)^n=a^n b^n\)
- \(a^0=1\;(a\ne0)\), \(a^{-n}=1/a^n\)
- \(a^{m/n}=\sqrt[n]{a^m}\)
Logs
- \(\log_b(MN)=\log_b M+\log_b N\)
- \(\log_b(M/N)=\log_b M-\log_b N\)
- \(\log_b(M^k)=k\log_b M\)
- \(\log_b x=\dfrac{\ln x}{\ln b}\)
Made for quick revision. Toggle solutions to self-check, or print the page for a one-page cheatsheet.
No comments:
Post a Comment