BeginnerTopic 9 of 104Engineering Mathematics
Exponents and logarithms
Logs invert exponentials and turn products into sums; essential for decibels, pH and growth laws.
In depth
The logarithm log_b(x) is the exponent to which b must be raised to give x. Key rules: log(ab)=log a+log b, log(a/b)=log a-log b, log(aⁿ)=n log a. Natural logs use base e≈2.718, common logs base 10. Decibels and pH are logarithmic scales.
Key points
- ln and eˣ are inverses
- Log scales compress huge ranges
- Change of base: log_b x = ln x / ln b
ExampleA 20 dB gain means a voltage ratio of 10, since dB=20·log10(ratio).
What is Exponents and logarithms?
Logs invert exponentials and turn products into sums; essential for decibels, pH and growth laws.
Why is it important to understand exponents and logarithms?
Logarithms compress huge ranges into manageable scales and turn multiplication into addition, which is why decibels and growth models use them.
Why are decibels logarithmic?
Human hearing and most sensors respond to ratios of intensity, so a log scale matches perception and allows large ranges to be written in small numbers.
How do I solve an equation with an unknown exponent?
Take the logarithm of both sides, use the power rule to bring the exponent down, then solve the linear result.
How is Exponents and logarithms used in practice?
Typical applications include expressing a power gain in decibels, calculating the pH of a solution and finding the time for a battery to discharge with exponential decay.
What should I learn before Exponents and logarithms?
Study "Quadratic equations" first (topic 8 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Exponents and logarithms?
ln and eˣ are inverses. Log scales compress huge ranges. Change of base: log_b x = ln x / ln b.
Can you give an example of Exponents and logarithms?
A 20 dB gain means a voltage ratio of 10, since dB=20·log10(ratio).
What comes after Exponents and logarithms?
The next topic is "Polynomials and factoring", which builds directly on this one.
How difficult is Exponents and logarithms?
It is classed as Beginner in Engineering Mathematics (topic 9 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.