Solve: a) `log_2.17(5x-1)−log_2.17(x+7)=0` b) `2e^x+2=5` c) `5+2ln(x)=4`

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lemjay | High School Teacher | (Level 3) Senior Educator

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(a) `log_2.17(5x-1)-log_2.17(x+7)=0`

Since the two logarithms have same base, express the right side as one logarithm.To do so, apply the quotient rule` log_b(M/N)=log_bM-log_bN` .


Then, convert to it to exponential equation. The equivalent exponential equation of `log_b M=a` is
`b^a=M` .



Now that the equation has no more logarithm, proceed to solve for x.







Hence, the solution to the given equation is `x=2` .

(b) `2e^x+2=5`

First, isolate e^x.





To remove the x in the exponent, take the logarithm of both sides.


`x ln e=ln(3/2)`


Hence, the solution to the equation is `x=ln (3/2)` .

(c) `5+2ln(x)=4`

First, isolate lnx.




Then, convert to its equivalent exponential equation.



Hence, the solution to the equation is `x=1/e^(1/2)` .