The Properties of Logarithms allow expressions with logarithms to be rewritten.
Simplify the expression log6(8)−log6(3)+log6(9)−log6(4) using the properties of logarithms.
The Change of Base Formula allows the logarithm of an arbitrary base to be rewritten as the quotient of two logarithms with another base. logc(a)=logb(c)logb(a) With many calculators it is only possible to evaluate the common and the natural logarithm. The Change of Base Formula can then be used to evaluate logarithms of other bases.
logc(a)=log(c)log(a)andlogc(a)=ln(c)ln(a)First, the equation is rewritten by applying a logarithm on both sides. 8x=3⇔log(8x)=log(3)
By using the Power Property of Logarithms, powers can be rewritten into a product. log(8x)=log(3)⇕x⋅log(8)=log(3)
After the power has been rewritten into a product, the unknown variable can be isolated using inverse operations. Here, x gets isolated on the left-hand side when both sides of the equation are divided by log(8). x⋅log(8)=log(3)⇕x=log(8)log(3) By using a calculator, an approximate value of x can be calculated. Here, x≈0.53.
Solve the equation 4x=25 using the common logarithm. State the answer with three significant digits.
A logarithmic function g(x)=logb(x), is by definition the inverse of an exponential function f(x)=bx. This means that their function composition results in the identity function.
The fact that a logarithm and a power with the same base undo
each other is what is known as the inverse properties of logarithms.
logb(bx)=xandblogb(x)=x
They also hold true for the common logarithm and the natural logarithm.
log(10x)=xandln(ex)=xand10log(x)=xeln(x)=x
These properties together with other properties of logarithms permit to simplify logarithmic expressions and to solve equations involving logarithms and powers. Some particular examples are shown below.
ln(e4x)4xx=20=20=53log3(5x)5xx=10=10=2
Solve the equation log5(125)eln(34)⋅10log(x)=log(100) using the inverse properties of logarithms.