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
$\log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n$
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.
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:
If $10^x=4$, find the value of $10^{2x+1}$.
We are given:
$$10^x = 4.$$
Rewrite 102x+1 using the product rule.
$$10^{2x+1} = 10^{2x} \cdot 10^1.$$
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, $$4^{6x-2y} = 4^y \;\;\Rightarrow\;\; 6x - 2y = y \;\;\Rightarrow\;\; 6x = 3y \;\;\Rightarrow\;\; y=2x.$$
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.
Which of the following can be a product of an even prime number and odd prime number.
4
12
6
8
The only even prime number is 2. Multiplying it by an odd prime gives a number of the form $2p$, where $p$ is an odd prime. Using $p = 3$: $2(3) = 6$ $\boxed{6}$
The product of $x$ and $y$ where $x < 0$ and $y \neq 0$.
negative
negative or positive
positive
can not be determined
Part 1.
The product of two negative numbers is positive, and multiplying that result by another negative number gives a negative number. $(-)(-)(-) = (+)(-) = -$ $\boxed{\text{negative}}$
Part 2.
A negative number divided by a positive number always has a negative sign. $(-) \div (+) = -$ $\boxed{\text{negative}}$
Part 3.
Since $x < 0$ but $y \neq 0$, the sign of $xy$ depends on the sign of $y$. If $y > 0$, then $xy < 0$. If $y < 0$, then $xy > 0$. $\boxed{\text{negative or positive}}$
Let "$a$" and "$b$" be ranges of numbers in a number line such that $-1 \leq a \leq 5$ and $6 \leq b \leq 10$. If "$a$" is shifted 6 units to the right and "$b$" is shifted 2 units to the right, how many common units will the shifted "$a$" and "$b$" share?
2
3
1
4
Shift range $a$ six units to the right: $-1 \leq a \leq 5 \Rightarrow 5 \leq a \leq 11$ Shift range $b$ two units to the right: $6 \leq b \leq 10 \Rightarrow 8 \leq b \leq 12$ The common interval is from 8 to 11, so the shared length is: $11 - 8 = 3$ $\boxed{3}$
How many rational roots have the following function? $f(x) = x^5 - 5x^4 + 5x^3 + 15x^2 - 36x + 20$.
1
2
3
0
How many possible positive real roots have the following function? $f(x) = 7x^7 + 5x^6 + 3x^3 + x$.
0
1
2
3
In the function $f(x) = x^{2n} - 1$, assuming $n$ is a natural number, how many are the positive roots?
2
1
0
3
Part 1.
Using the Rational Root Theorem, test the possible rational factors of the constant term 20. The function has rational zeros at: $x = -2,\;1,\;2$ Thus the number of rational roots is: $\boxed{3}$
Part 2.
For $f(x) = 7x^7 + 5x^6 + 3x^3 + x$, all coefficients are positive. There is no sign change in $f(x)$, so by Descartes' Rule of Signs there are no positive real roots. $\boxed{0}$
Part 3.
Solve $x^{2n} - 1 = 0$: $x^{2n} = 1$ For natural number $n$, the positive real solution is $x = 1$. $\boxed{1}$
A poultry farm has only chickens and pigs. When the manager of the poultry counted the heads of the stock in the farm, the number totaled up to 200. However, when the number of legs was counted, the number totaled up to 540. How many chickens were there in the farm?
70
130
120
80
Let $c$ be chickens and $p$ be pigs. $c+p=200$ $2c+4p=540$ From $c=200-p$: $2(200-p)+4p=540$ $400+2p=540 \Rightarrow p=70$, so $c=130$. $\boxed{130}$
Mario's hat is four more than Alex's hat and one-half that of Miguel's hat. If the total number of hats is 24, how many hats does Miguel have?
14
3
7
16
Let Alex have $a$ hats. Then Mario has $a+4$. Mario is one-half of Miguel, so Miguel has $2(a+4)$. $a+(a+4)+2(a+4)=24$ $4a+12=24 \Rightarrow a=3$ Miguel has $2(3+4)=14$ hats. $\boxed{14}$
Convert the following complex number to rectangular form: $480\angle-135^\circ$.
$-339.4 - 339.4i$
$-339.4 + 339.4i$
$339.4 - 339.4i$
$339.4 + 339.4i$
Convert polar to rectangular form using $r(\cos\theta+i\sin\theta)$. $480\angle-135^\circ=480(\cos(-135^\circ)+i\sin(-135^\circ))$ $=480\left(-\frac{\sqrt2}{2}-i\frac{\sqrt2}{2}\right)$ $=-339.4-339.4i$. $\boxed{-339.4-339.4i}$
Nine subtracted from 8 times a number is 39. What is the number?
5
8
7
6
Solution pending in psadquestions/t1372.json.
Question Bank: t2111
MSTE - Algebra / Exponents and Logarithms / Besavilla CE Pre-Board Math & Surveying
Solve for $b$ from the given equation: $2\log b^2 - 3\log b = \log 8b - \log 4b$
1
2
3
4
5
For real logarithms, take $b>0$. Apply log laws: $2\log b^2=\log b^4$ and $3\log b=\log b^3$ Left side: $\log b^4-\log b^3=\log b$ Right side: $\log 8b-\log 4b=\log\left(\frac{8b}{4b}\right)=\log2$ Thus $\log b=\log2$, so $\boxed{b=2}$
Question Bank: t2152
MSTE - Algebra / Exponents and Logarithms / Besavilla CE Pre-Board Math & Surveying
Solve for $x$ from the equation: $\log(x - 1) + \log(x + 1) = 2\log(x + 2)$
- 7/4
- 3/4
- 5/4
- 9/4
- 1/4
Using log laws on the printed equation gives $\log[(x-1)(x+1)]=\log[(x+2)^2]$ $(x-1)(x+1)=(x+2)^2$ $x^2-1=x^2+4x+4 \Rightarrow -5=4x$ $x=-\frac{5}{4}$ Note that for real logarithms in the printed equation, $x-1$ and $x+1$ must be positive, so this root is outside the usual domain. The keyed algebraic answer is $\boxed{-\frac{5}{4}}$.
