# Properties of Logarithms and How to Solve Log Equations

In mathematics, logarithms are the inverses of exponential functions. That is, if b^{x} = y, then x = Log_{b}(y), where the notation Log_{b} means logarithm in base b. Conventionally, "Log" or "log" denotes the logarithm base 10 and "Ln" or "ln" denotes base e = 2.7182818... Logs are used to solve equations involving exponents, and conversely, exponents are used to solve equations involving logs.

There are several properties of logarithms that will help you simplify and solve equations. The following seven rules are true regardless of the base. Here, "log" denotes logarithm with arbitrary base unless otherwise noted:

(1) log(a) + log(b) = log(ab)

(2) log(a) - log(b) = log(a/b)

(3) log(a^{n}) = n*log(a)

(4) log(1) = 0

(5) m^[log_{m}(a)] = a

(6) log_{m}(a) = log(a)/log(m)

(7) b^[log(a)] = a^[log(b)]

Additionally, Log(10) = 1 and Ln(e) = 1.

### Some Examples of Solving Log Equations

**Example 1:**Solve 2Log(x) + Log(7) = Log(x) + 4, where Log is the base 10 logarithm.

2Log(x) + Log(7) = Log(x) + 4

2Log(x) - Log(x) = 4 - Log(7)

Log(x) = 4 - Log(7)

10^[Log(x)] = 10^[4 - Log(7)]

x = 10

^{4}/10

^{Log(7)}

x = 10000/7

**Example 2:**Solve 2Ln(x) - Ln(x-3) = Ln(14).

2Ln(x) - Ln(x-3) = Ln(14)

Ln(x

^{2}) - Ln(x-3) = Ln(14)

Ln[x

^{2}/(x-3)] = Ln(14)

e^Ln[x

^{2}/(x-3)] = e^Ln(14)

x

^{2}/(x-3) = 14

x

^{2}= 14(x-3)

x

^{2}- 14x + 42 = 0

x = 9.646, 4.354

**Example 3:**Solve log

_{3}(4x) = log

_{4}(3x).

log

_{3}(4x) = log

_{4}(3x)

Ln(4x)/Ln(3) = Ln(3x)/Ln(4)

[Ln(4) + Ln(x)]/Ln(3) = [Ln(3) + Ln(x)]/Ln(4)

Ln(4)*Ln(4) + Ln(4)*Ln(x) = Ln(3)*Ln(3) + Ln(3)*Ln(x)

Ln(4)*Ln(x) - Ln(3)*Ln(x) = Ln(3)*Ln(3) - Ln(4)*Ln(4)

Ln(x)*[Ln(4) - Ln(3)] = Ln(3)*Ln(3) - Ln(4)*Ln(4)

Ln(x) = [Ln(3)

^{2}- Ln(4)

^{2}]/[Ln(4) - Ln(3)]

Ln(x) = -Ln(3) - Ln(4) = -Ln(12)

x = 1/12

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