Solve the logarithmic equation. Express irrational solutions in exact form and a

sanuluy

sanuluy

Answered question

2021-11-07

Solve the logarithmic equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places.
2log5(x3)log58=log52

Answer & Explanation

Aniqa O'Neill

Aniqa O'Neill

Skilled2021-11-08Added 100 answers

Step 1
Given logarithmic equation in x can be expressed as a quadratic equation in x without the logarithms. This is done using properties of logarithm. One property that will be used is mlogab=logabm.
Another property that will be used is logaxlogay=logaxy. Another property of logarithm function that will be used is that logarithm functions are one-to-one, so if we have logx=log y then it implies x=y. One point to note is that argument of a logarithm should be positive so check for each solution if log functions in the equation are defined for each solution.
Step 2
Given equation is 2log5(x3)log58=log52. Use the property of logarithms to convert this into a quadratic in x and solve for x.
2log5(x3)log58=log52
log5(x3)2log58=log52
log5(x3)28=log52
(x3)28=2
(x3)2=16
x3=4,4
x=1,7
Now for x=−1 the argument of log5(x−3) will be negative which is not allowed for logarithmic functions.
Hence, the solution to the given equation is x=7.

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