Logarithm Change Of Base Rule

Change of Base of operations Formula

The alter of base formula, as its name suggests, is used to change the base of a logarithm. We might take noticed that a scientific figurer has only "log" and "ln" buttons. Also, nosotros know that "log" stands for a logarithm of base x and "ln" stands for a logarithm of base e. Merely there is no option to calculate the logarithm of a number with whatever other bases other than 10 and e.

The modify of base formula solves this issue of changing base from east to x, and from base ten to e. Also, it is used in solving several logarithms bug. Allow usa acquire the change of base formula forth with its proof and a few solved examples.

What Is Change of Base Formula?

The alter of base formula is used to write a logarithm of a number with a given base of operations every bit the ratio of two logarithms each with the aforementioned base that is different from the base of the original logarithm. This is a belongings of logarithms. You can see the change of base of operations formula here.

Change of base formula

Modify of Base of operations Formula

Alter of base formula tin can be represented equally follows. Here the base of the given logarithm is likewise changed to a logarithm with a new base. The basic logarithm with a base is transformed to two logarithms with a new and same base. The modify of base formula is:

log\(_b\) a = [log\(_c\) a] / [log\(_c\) b]

In this formula,

The argument of the logarithm in the numerator is the same every bit the argument of the original logarithm.
The argument of the logarithm in the denominator is the same as the base of the original logarithm.
The bases of both logarithms of numerator and denominator should be the same and this base tin can be whatever positive number other than 1.

Annotation:Another class of this formula is, log\(_b\) a · log\(_c\) b = log\(_c\) a, which is also widely used in solving the bug.

Change of Base Formula Derivation

Change of base formula derivation can exist understood from the post-obit steps. Let us presume that

log\(_b\) a = p,log\(_c\) a = q, and log\(_c\) b = r.

By converting each of these into exponential grade, we go,

a = b^p, a = c^q, and b = c^r.

From the kickoff two equations,

b^p = c^q

Substituting b = c^r (which is from third equation) hither,

(c^r)^p= c^q

c^rp = c^q (Using a property of exponents, (a^m)ⁿ = a^mn)

pr = q

p = q / r

Substituting the values of p, q, and r hither,

log\(_b\) a = [log\(_c\) a] / [log\(_c\) b]

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Let u.s.a. see the applications of the change of base of operations formula in the post-obit department.

Examples Using Change of Base Formula

Example ane:Evaluate the value of log\(_{64}\) 8 using the change of base formula.

Solution:

We volition apply the change of base formula (by changing the base to 10). Note that log\(_{10}\) is same as log.

log\(_{64}\) 8 = [log 8] / [log 64]

= [log 8] / [log 8²]

= [log viii] / [two log viii] [∵ log a^k = m log a]

= ane / two

Answer:log\(_{64}\) 8 = 1 / 2.

Example 2:Caluclate log\(_9\) eight using the figurer. Round your answer to four decimals.

Solution:

Nosotros cannot calculate log\(_9\) viii directly using the calculator because information technology doesn't accept a button named log\(_9\). Thus, we apply the modify of base formula first.

log\(_9\) 8 = [log eight] / [log 9]

= [0.903089...] / [0.95424...]

≈ 0.9464

Answer:log\(_9\) 8 ≈ 0.9464.

Instance iii:Evaluate the value of log\(_3\) 2 · log\(_4\) 3 · log\(_5\) iv.

Solution:

By alternating form of the change of base of operations formula, log\(_b\) a ⋅ log\(_c\) b = log\(_c\) a. We apply this twice to evaluate the given expression.

log\(_3\) 2 · log\(_4\) 3 · log\(_5\) 4

= log\(_4\) two · log\(_5\) 4

= log\(_5\) two

Reply:log\(_3\) 2 · log\(_4\) 3 · log\(_5\) 4 = log\(_5\) 2.

FAQs on Alter of Base of operations Formula

What Is Change of Base of operations Formula?

The change of base formula is used to change the base of a logarithm. It has two forms.

log\(_b\) a = [log\(_c\) a] / [log\(_c\) b]
log\(_b\) a ⋅ log\(_c\) b = log\(_c\) a

How To Derive Change of Base Formula?

The change of base of operations formula says, log\(_b\) a = [log\(_c\) a] / [log\(_c\) b]. To prove this, we presume that log\(_b\) a = p,log\(_c\) a = q, and log\(_c\) b = r. Converting each of these into the exponential form, nosotros become a = b^p, a = c^q, and b = c^r. From the first two equations, b^p = c^q. Substituting b = c^r (which is from 3rd equation) hither,

(c^r)^p= c^q

c^rp = c^q

pr = q

p = q / r

(or) log\(_b\) a = [log\(_c\) a] / [log\(_c\) b]

What Are the Applications of Change of Base Formula?

The change of base formula is mainly used to modify the base of operations of a logarithm to any desired base. This is many used to summate the logarithms with any other base than x and "e" because the estimator has options to calculate the logarithms with bases 10 (log push) and e (ln push button) only.

How To Use Alter of Base Formula?

The change of base formula says log\(_b\) a = [log\(_c\) a] / [log\(_c\) b]. It means to change the base of a logarithm log\(_b\) a, we merely use division [log a] / [log b] where these logarithms can accept any (same) positive number as a base.