Question

What is the exact solution

Answer 1

Answer:

$\displaystyle{x = \dfrac{\ln 5+6 \ln 10}{2\ln 10} }$

Step-by-step 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 $\displaystyle{\ln 10}$:

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

Add 6 both sides:

$\displaystyle{2x-6+6 = \dfrac{\ln 5}{\ln 10}+6}\\\\\displaystyle{2x = \dfrac{\ln 5}{\ln 10}+6}$

Then divide both sides by 2:

$\displaystyle{\dfrac{2x}{2} = \dfrac{\dfrac{\ln 5}{\ln 10}+6}{2}}\\\\\displaystyle{x = \dfrac{\ln 5}{2\ln 10} + 3}$

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

$\displaystyle{x = \dfrac{\ln 5}{2\ln 10} + \dfrac{3 \cdot 2\ln 10}{1 \cdot 2\ln 10}}\\\\\displaystyle{x = \dfrac{\ln 5}{2\ln 10} + \dfrac{6\ln 10}{ 2\ln 10}}$

Add in one denominator:

$\displaystyle{x = \dfrac{\ln 5+6 \ln 10}{2\ln 10} }$

Hence, the answer is first choice.

Answer 2

Step-by-step 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.