Question

What is the exact solution

Answer 1

Explanation:

10^(2x - 6) = 5

i would use immediately a log10 of both sides and get

2x - 6 = log10(5)

since the answers want to use ln, let's convert the log10 term to ln (rules of logarithm - change of base rule) :

log10(5) = ln(5)/ln(10)

so, we have

2x - 6 = ln(5)/ln(10)

then

2x = ln(5)/ln(10) + 6

x = ln(5)/(2ln(10)) + 6/2

and now to bring the whole right side into one fraction, we need to transform 6/2 into something like .../(2ln(10)).

so, we get

x = ln(5)/(2ln(10)) + (6ln(10))/(2ln(10)) =

= (ln(5) + 6ln(10))/(2ln(10))

that is the first answer option.

Answer 2

`\displaystyle{x = (\ln 5+6 \ln 10)/(2\ln 10) }`

Explanation:

We are given the exponential equation of:

`\displaystyle{10^(2x-6) = 5}`

Take natural logarithm both sides:

`\displaystyle{\ln \left(10^(2x-6)\right) = \ln 5}`

Apply logarithm property where:

`\displaystyle{\ln M^N = N\ln M}`

Therefore:

`\displaystyle{(2x-6)\ln 10 = \ln 5}`

Divide both sides by
`\displaystyle{\ln 10}`:

`\displaystyle{((2x-6)\ln 10)/(\ln 10) = (\ln 5)/(\ln 10)}\\\\\displaystyle{2x-6 = (\ln 5)/(\ln 10)}`

Add 6 both sides:

`\displaystyle{2x-6+6 = (\ln 5)/(\ln 10)+6}\\\\\displaystyle{2x = (\ln 5)/(\ln 10)+6}`

Then divide both sides by 2:

`\displaystyle{(2x)/(2) = ((\ln 5)/(\ln 10)+6)/(2)}\\\\\displaystyle{x = (\ln 5)/(2\ln 10) + 3}`

Multiply 3 by 2ln10 for both denominator (1) and numerator (3):

`\displaystyle{x = (\ln 5)/(2\ln 10) + (3 \cdot 2\ln 10)/(1 \cdot 2\ln 10)}\\\\\displaystyle{x = (\ln 5)/(2\ln 10) + (6\ln 10)/( 2\ln 10)}`

Add in one denominator:

`\displaystyle{x = (\ln 5+6 \ln 10)/(2\ln 10) }`

Hence, the answer is first choice.