log(2,x)=3 — step-by-step solution

Let's solve in

ℝ

\log (2, x) = 3

ⓘ log(b, v) is the logarithm of v in base b, that is ln(v)/ln(b):

\log (2, x)

\frac{\ln (x)}{\ln (2)}

\frac{\ln (x)}{\ln (2)} = 3

\ln (x) = 3 \cdot \ln (2)

ⓘ We subtract

3 \cdot \ln (2)

from both sides

\ln (x) - 3 \cdot \ln (2) = 0

ⓘ For strictly positive arguments, a sum of logarithms is the logarithm of the product, a difference that of the quotient, ln(a) + ln(b) = ln(a*b):

\ln (x) - 3 \cdot \ln (2)

\ln (\frac{x}{2^{3}})

\ln (\frac{x}{2^{3}}) = 0

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

2^{3}

2 \cdot 2 \cdot 2

8

\ln (\frac{x}{8}) = 0

You can apply the exp function on both sides:

e^{\ln (\frac{x}{8})} = e^{0}

ⓘ For real

x

\ln (\exp (x)) = x

; wherever

\ln (x)

is defined,

\exp (\ln (x)) = x

e^{\ln (\frac{x}{8})}

\frac{x}{8}

■Any nonzero number raised to the power 0 is 1:

e^{0}

1

\frac{x}{8} = 1

Candidates, pending the domain check:

x = 8

For the expression

\ln (x)

to exist, you must check:

x > 0

Domain constraints:

x > 0

▶Solution:

x = 8