Exponents and Logarithms

Logarithms and exponents are inverse operations. These properties allow us to simplify, expand, or solve exponential and logarithmic expressions, especially in engineering applications such as signal processing, decay models, and differential equations.

Rule Name	Property
Log of 1	$\log_b (1) = 0$
Log of the same number as base	$\log_b (b) = 1$
Product Rule	$\log_b(m\cdot n) = \log_b (m) + \log_b (n)$
Quotient Rule	$\log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n$
Power Rule	$\log_b (m)^n = n\cdot \log_b (m)$
Change of Base Rule	$\log_a (b) = \frac{\log_c b}{\log_c a}$ (or) $\log_a (b) \cdot \log_c a = \log_c b$
Equality Rule	$\log_a (b) = \log_c (b) \Rightarrow a = c$
Number Raised to Log	$b^{\log_b x} = x$
Other Rules	$\log_a \left( a^m \right) = m$ $- \log_b a = \log_b \left( \frac{1}{a} \right)$ (or) $\log_b \left( \frac{1}{a} \right) = -\log_b a$

Exponent Rules

Exponents follow specific rules that simplify algebraic manipulation in engineering and mathematics. These rules help when working with growth models, circuits, or solving higher-order equations.

Rule Name	Property
Product Rule	$a^m \cdot a^n = a^{m+n}$
Quotient Rule	$\dfrac{a^m}{a^n} = a^{m-n}$
Power of a Power	$(a^m)^n = a^{mn}$
Power of a Product	$(ab)^n = a^n \cdot b^n$
Power of a Quotient	$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$
Zero Exponent	$a^0 = 1 \quad (a \neq 0)$
Negative Exponent	$a^{-n} = \dfrac{1}{a^n}$
Negative Exponent with Fraction	$\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$

Natural Logarithm (ln) and Euler’s Number

Euler’s number $e \approx 2.718$ is the base of the natural logarithm. When working with exponential growth, decay, or continuous compounding, we use:

$$ \ln x = \log_e x $$

This is especially useful when solving equations involving exponential functions like $e^x$, and appears frequently in calculus, differential equations, and engineering analysis.

Common Logarithm (log)

In most scientific and engineering calculators, when you enter log without specifying a base, it is understood to be base 10 by default. This is known as the common logarithm.

$$ \log x \equiv \log_{10} x $$

This is especially useful in engineering fields where base-10 logarithms are standard for expressing quantities that span multiple orders of magnitude, such as decibels (dB), pH values, or Richter scale readings.

Problem 1:

If $10^x=4$, find the value of $10^{2x+1}$.

We are given: $$10^x = 4.$$
Rewrite 10^2x+1 using the product rule. $$10^{2x+1} = 10^{2x} \cdot 10^1.$$
Rewrite $10^{2x}$ as: $$(10^x)^2.$$
Substitute $10^x = 4$: $$(10^x)^2 \cdot 10 = (4)^2 \cdot 10.$$
Compute: $$16 \cdot 10 = 160.$$

Final Answer: $\boxed{10^{2x+1} = 160}$

Problem 2:

Determine the value of x and y if $8^x=2^{y+2}$ and $16^{3x-y}=4^y$

Rewrite the first equation in terms of base $2$: $$8^x = (2^3)^x = 2^{3x}.$$ So, $$2^{3x} = 2^{y+2} \;\;\Rightarrow\;\; 3x = y+2 \;\;\Rightarrow\;\; y = 3x-2.$$

For the second equation, rewrite in terms of base $4$: $$16^{3x-y} = (4^2)^{3x-y} = 4^{2(3x-y)} = 4^{6x-2y}.$$ So, $$4^{6x-2y} = 4^y \;\;\Rightarrow\;\; 6x - 2y = y \;\;\Rightarrow\;\; 6x = 3y \;\;\Rightarrow\;\; y=2x.$$

Substitute $y=2x$ into $y=3x-2$: $$2x = 3x - 2 \;\;\Rightarrow\;\; x=2.$$

Find $y$: $$y=2x=2(2)=4.$$

Final Answer: $\boxed{x=2,\; y=4}$

Problem 3:

If $2log_4x - log_49=2$, find x.

Power rule: $$2\log_4 x=\log_4 x^2.$$
Quotient rule: $$\log_4 x^2-\log_4 9=\log_4\!\left(\frac{x^2}{9}\right)=2.$$
Number–raised–to–log rule $b^{\log_b x}=x$: $$\boxed{\,4^{\log_4\left(\frac{x^2}{9}\right)} = 4^{\,2}\,}\;\Rightarrow\; \frac{x^2}{9}=16.$$
Then $x^2=144 \Rightarrow x=12$ (domain $x>0$) since x is inside a log.

Final Answer: $\boxed{x=12}$

In the real number system, there is no logarithm of a negative value. However, in complex numbers, we can extend logarithms to negative values using Euler's formula.

In the complex domain (principal value):

$$\log(-A)=\ln(A)+i\pi \quad (A>0).$$

General multivalued form:

$$\log(-A)=\ln(A)+i(2k+1)\pi,\;\; k\in\mathbb{Z},\; A>0.$$