Laws of logarithms are rules that can be applied to simplify and solve complicated logarithmic equations. When manipulating logarithms, it is important to make sure all the bases are the same.
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Product (addition) Law: \(\log_a(m) + \log_b(n) = \log_a(mn)\)
Quotient Law: \(\log_a(m) - \log_b(n) = \log_a(\frac{m}{ n})\)
Power Law: \(\log_a(x^b) = b\log_a(x)\)
Change of Law: \(\log_a(x) = \frac{\log_b(x)}{\log_b(a)}\)
The change of law formula is in the formula booklet you are given in the exam.
Other logarithms
Reciprocal Laws: \(\log_a(\frac{1}{x}) = \log(x^{-1}) = -\log(x)\)
Log of the base: \(\log_a(a)=1\)
Log of 1: \(\log_a(1) = 0\)
Although technically a logarithm law, it is important to remember logarithms can be converted into exponentials: \(\log_a(b) = x\) can be written as \(a^x = b\).
It is not necessary to be able to prove each logarithm law for the exam, but it is important to understand each step and why it occurs.
1. If \(\log_x(a) = c\) and \(\log_x(b) = d\), then you can rewrite the logarithms as an exponential function.
For \(\log_x(a) = c\), the base isx, the exponent is c, the answer to the exponential is a.
Therefore, it can be written as \(x^c = a\)
For \(\log_x(b) = d\), the base is x, the exponent is d, and the answer to the exponential is b.
Therefore, it can be written as \(x^d = b\)
2. Thus, using our exponential (indices) rule of \(M^n \cdot M^n = M^{m+n}\),
\(ab = (x^c)(x^d) = x^{c+d}\)
\(ab = x^{c+d}\)
3. Take the log of both sides:
\(\log_x(ab) = \log_x(x^{c+d})\)
4. Because \(\log_x(x^{c+d})\) includes both an exponential with a base of x and a logarithm with a base of x (\(\log_x(x^{c+d})\)), they will cancel each other out to become just c + d.
\(log_x(x^{c+d}) = c +d \)
\(\log_x(ab) = c+d\)
This step is because logarithms and exponentials are inverse functions of each other. Think about when we cancel out the +4 and -4 in x +4 -4 = 10 – this is the same principle.
5. We defined c and d in part 1. \(\log_x(a) = c\) and \(\log_x(b) = d\)
Therefore, \(c +d = \log_x(a) + \log_x(b)\)
\(\log_x(ab) = \log_x(a) + \log_x(b)\)
1. If \(\log_x(a) = c\) and \(\log_x(b) = d\), then you can rewrite the logarithms as an exponential function.
For\(\log_x(a) = c\), the base is x, the exponent is c, and the answer to the exponential is a.
Therefore, it can be written as \(x^c = a\)
For \(\log_x(b) = d\), the base is x, the exponent is d, and the answer to the exponential is b.
Therefore, it can be written as \(x^d = b\)
2. Thus, using our exponential (indices) rules of \(\frac{M^m}{M^n} = M^{m-n}\),
\(\frac{a}{b} = \frac{x^c}{x^d} = x^{c-d}\)
\(\frac{a}{b} = x^{c-d}\)
3. Take the log of both sides.
\(\log_x(\frac{a}{b}) = \log_x(x^{c-d})\)
4. Because \(\log_x(x^{c-d})\) includes both an exponential with a base of x and a logarithm with a base of x, they will cancel each other out to become just c - d .
\(\log_x(x^{c-d}) = c-d\)
\(\log_x(\frac{a}{b}) = c-d\)
5. We defined c and d in part 1 where \(\log_x(a) = c\) and \(\log_x(b) = d\):
\(c-d = \log_x(a) - \log_x(b)\)
\(\log_x(\frac{a}{b}) = \log_x(a) - \log_x(b)\)
1. Let \(\log_a(x) = k\) where the base is a ,the exponent is k, and the answer to the exponential = x.
Therefore, it can be rewritten as an exponential: \(a^k = x\)
2. Take the log of both sides
\(\log_b(a^k) = \log_b(x)\)
3. Use the power rule to simplify
\(\log_b(a^k) = k\log_b(a)\) which you can then substitute back into the equation
\(k\log_b(a) = \log_b(x)\)
4. Rearrange to get k on its own by dividing through k \(\log_b(a)\)
\(k = \frac{\log_b(x)}{\log_b(a)}\)
5. As k is already defined, it can be substituted into the equationk\(\log_a(x)\)
\(\log_a(x) = \frac{\log_b(x)}{\log_b(a)}\)
Here, we will go through some examples of simplifying a range of laws of logs.
Show \(\log (6) + \log (4) = log (24)\)
\(\log (6) + \log (4) = \log (6 \cdot 4) = \log (24)\)
Solve \(\log (14) - \log (7)\)
\(\log (14) - \log (7) = \log(\frac{14}{7}) = \log (2) = 0.301 (3 s.f)\)
Simplify \(2\log (9)\), keep in exact form
\(2\log(9) = \log(9)^2 = \log(81)\)
Solve \(2\log(2\cdot 3)\)
\(2\log (2 \cdot 3) = \log(2 \cdot 3)^2 = \log(6)^2 = \log(36) = 1.56 (3 s.f)\)
It might help to use rules that simplify individual logs before doing the simplifying multiple log laws.
Solve \(3\log(4) - \log(8)\)
\(\log(4)^3 = \log(64)\log(64) - \log(8) = \log(\frac{64}{8}) = \log(8) = 0.903 (3 s.f)\)Simplify \(\log_4(4x^2)\)
\(\log_4(4) + \log_4(x^2)1+ 2\log_4x\)
Prove \(x = 1 \pm i\sqrt{8}\) where \(2\log_2(x+3) - log_2(x) = 3\)
1. Using the power rule, \(2\log_2(x+3) = \log_2(x+3)^2\).
Therefore, \(\log_2(x+3)^2 - \log_2(x) = 3\)
2. Using the quotient rule, \(\frac{\log_2(x+3)^2}{\log_2(x)} = 3\)
3. When you want to remove the logarithm, you need to convert it into an exponential. This works in the same way as normal – just make sure you label each part.
Base is 2; exponent is 3; answer to exponential is \(\frac{(x+3)^2}{x}\)
\(2^3 = \frac{(x+3)^2}{x}\)
4. Solve like a normal equation
\(8 = \frac{(x+3)^2}{x}\)
\(8x = (x+3)^2\)
\(0 = x^2 -2x+9\)
Using the formula we get, \(x = 1 \pm i\sqrt{8}\)
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What are the laws of logarithms?
The laws of logarithms are the rules that you can use to simplify and solve complicated logarithmic equations.
What are the four laws of logarithms?
There are four main types of logs of logarithms: product (addition), quotient, change of base, and the power law.
How do you change the base of a logarithm?
You can change the base of a logarithm by doing loga(b)/logb(x)
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