Solve the Logarithmic Equation: log₇(x⁴) - log₇(2x²) = 3

log⁡7x4−log⁡72x2=3 \log_7x^4-\log_72x^2=3

?=x

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Step-by-step video solution

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00:00 Solve
00:05 We'll use the logarithm subtraction formula, we'll get the logarithm of their quotient
00:12 We'll use this formula in our exercise
00:26 Let's simplify what we can
00:37 Solve according to the logarithm definition
00:45 Isolate X
01:11 When extracting a root there are always 2 solutions, positive and negative
01:16 These are the possible solutions
01:29 Let's check the domain
01:34 According to the domain we'll find the solution
01:41 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

log⁡7x4−log⁡72x2=3 \log_7x^4-\log_72x^2=3

?=x

2

Step-by-step solution

log⁡ax−log⁡ay=log⁡axy \log_ax-\log_ay=\log_a\frac{x}{y}

log⁡7x4−log⁡72x2= \log_7x^4-\log_72x^2=

log⁡7x42x2=3 \log_7\frac{x^4}{2x^2}=3

73=x22 7^3=\frac{x^2}{2}

We multiply by: 2 2

2⋅73=x2 2\cdot7^3=x^2

Extract the root

x=680=714 x=\sqrt{680}=7\sqrt{14}

x=−680=−714 x=-\sqrt{680}=-7\sqrt{14}

3

Final Answer

−714  , 714 -7\sqrt{14\text{ }}\text{ , }7\sqrt{14}

Practice Quiz

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\( \log_{10}3+\log_{10}4= \)

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