melodykap
2021-08-03
Answered

Exponential and Logarithmic Equations Solve the equation. Find the exact solution if possible. otherwise, use a calculator to approximate to two decimals.

${\mathrm{log}}_{3}(x-8)+{\mathrm{log}}_{3}x=2$

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krolaniaN

Answered 2021-08-04
Author has **86** answers

Step 1

For solving Exponential Equations:

1) Isolate the exponential expression on one side of the equation.

2) Take the logarithm of each side, then use the Laws of Logarithms to "bring down exponent".

3) Solve for the variable.

Step 2

Given,

${\mathrm{log}}_{3}(x-8)+{\mathrm{log}}_{3}x=2$

By the first law of logarithms function,

${\mathrm{log}}_{3}(x-8)x=2$

By the definition of logarithm function,

${\mathrm{log}}_{a}x=y\iff \text{}{a}^{y}=x$

${3}^{2}=x(x-8)$

$9=x(x-8)$

By the distributive property of multiplication,

$x\cdot \text{}x-x\cdot 8=9$

${x}^{2}-8x=9$

Add$-9$ on both sides,

${x}^{2}-8x-9=9-9$

${x}^{2}-8x-9=0$

Factor${x}^{2}-8x-9$ as $(x-9)(x+1).$

$(x-9)(x+1)=0$

By the zero-factor property,

$x-9=0$ and $x+1=0$

Taking,

$x-9=0$

Add 9 on both sides,

$x-9+9=0+9$

$x=9$

Now taking,

$x+1=0$

Add$-1$ on both sides,

$x+1-1=0-1$

$x=-1$

Here, we neglect$x=-1$ because logarithm function cannot be negative.

Thus, the exact solution is$x=9.$

For solving Exponential Equations:

1) Isolate the exponential expression on one side of the equation.

2) Take the logarithm of each side, then use the Laws of Logarithms to "bring down exponent".

3) Solve for the variable.

Step 2

Given,

By the first law of logarithms function,

By the definition of logarithm function,

By the distributive property of multiplication,

Add

Factor

By the zero-factor property,

Taking,

Add 9 on both sides,

Now taking,

Add

Here, we neglect

Thus, the exact solution is

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The equation