Indices and Logarithms

Laws of Logarithms

Logarithms are the inverse operations of exponents. If:

$$\log_b x = y \quad \text{then} \quad b^y = x$$logb​x=ythenby=x

where $b$b is the base, $x$x is the number, and $y$y is the logarithm.

The fundamental laws of logarithms are as follows:

---

1. Product Law

$$\log_b (MN) = \log_b M + \log_b N$$logb​(MN)=logb​M+logb​N

Example:

$$\log_2 (8 \times 4) = \log_2 8 + \log_2 4$$log2​(8×4)=log2​8+log2​4

Since $\log_2 8 = 3$log2​8=3 and $\log_2 4 = 2$log2​4=2, we get:

$$\log_2 32 = 3 + 2 = 5$$log2​32=3+2=5

---

2. Quotient Law

$$\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N$$logb​(NM​)=logb​M−logb​N

Example:

$$\log_3 \left( \frac{81}{9} \right) = \log_3 81 - \log_3 9$$log3​(981​)=log3​81−log3​9

Since $\log_3 81 = 4$log3​81=4 and $\log_3 9 = 2$log3​9=2, we get:

$$\log_3 9 = 4 - 2 = 2$$log3​9=4−2=2

---

3. Power Law

$$\log_b (M^n) = n \log_b M$$logb​(Mn)=nlogb​M

Example:

$$\log_5 (25^3) = 3 \log_5 25$$log5​(253)=3log5​25

Since $\log_5 25 = 2$log5​25=2, we get:

$$\log_5 (25^3) = 3 \times 2 = 6$$log5​(253)=3×2=6

---

4. Change of Base Law

$$\log_b M = \frac{\log_k M}{\log_k b}$$logb​M=logk​blogk​M​

This is useful for converting logarithms into a different base, commonly base 10 (logarithm) or base $e$e (natural logarithm, denoted as $\ln$ln).

Example: Convert $\log_2 10$log2​10 to base 10.

$$\log_2 10 = \frac{\log_{10} 10}{\log_{10} 2}$$log2​10=log10​2log10​10​

Since $\log_{10} 10 = 1$log10​10=1 and $\log_{10} 2 \approx 0.301$log10​2≈0.301, we get:

$$\log_2 10 = \frac{1}{0.301} \approx 3.32$$log2​10=0.3011​≈3.32

---

5. Logarithm of 1

$$\log_b 1 = 0$$logb​1=0

Because any number raised to the power of zero equals 1:

$$b^0 = 1$$b0=1

Example:

$$\log_7 1 = 0$$log7​1=0

---

6. Logarithm of the Base

$$\log_b b = 1$$logb​b=1

Because $b^1 = b$b1=b.
Example:

$$\log_4 4 = 1$$log4​4=1

---

7. Natural Logarithms (Base $e$e)

$$\ln x = \log_e x$$lnx=loge​x

Natural logarithms use the base $e$e (Euler’s number, approximately 2.718). The same logarithmic laws apply:

$$\ln (MN) = \ln M + \ln N$$ln(MN)=lnM+lnN $$\ln \left( \frac{M}{N} \right) = \ln M - \ln N$$ln(NM​)=lnM−lnN $$\ln (M^n) = n \ln M$$ln(Mn)=nlnM

---

Examples Using Multiple Laws

1. Simplify $\log_3 81 + \log_3 9$log3​81+log3​9

   • Using the product law:

     $$\log_3 (81 \times 9) = \log_3 729$$log3​(81×9)=log3​729

   • Since $3^6 = 729$36=729, we get:

     $$\log_3 729 = 6$$log3​729=6

2. Simplify $\log_5 125 - \log_5 25$log5​125−log5​25

   • Using the quotient law:

     $$\log_5 \left( \frac{125}{25} \right) = \log_5 5$$log5​(25125​)=log5​5

   • Since $\log_5 5 = 1$log5​5=1, the answer is:

     $$1$$1

3. Simplify $2 \log_4 8$2log4​8

   • Using the power law:

     $$\log_4 8^2 = \log_4 64$$log4​82=log4​64

   • Since $4^3 = 64$43=64, we get:

     $$\log_4 64 = 3$$log4​64=3