Question

Log subscript-4 (m^2)=log subscript-4 (18-7m)?

Answer 1

`\displaystyle m_(1) = 2 \quad \text{and} \quad \displaystyle m _(2) = - 9`

Explanation:

we are given a logarithm equation

`\displaystyle \log_(4)( {m}^(2) ) = \log_(4)(18 - 7m)`

notice that, we have
`\log_4` both sides therefore we can get rid of it

`\displaystyle {m}^(2) = 18 - 7m`

in order to solve it we should make it standard form we know that

`\displaystyle a {x}^(2) + bx + c = 0`

so right hand side expression to left hand side and change its sign:

`\displaystyle {m}^(2) + 7m- 18=0`

now we can solve it by using factoring method

to do so rewrite the middle term as sum or subtraction of two different terms

in that case -2m+9m is good to use

`\displaystyle {m}^(2) - 2m + 9m - 18 = 0`

factor out m:

`\displaystyle m({m}^{} - 2 )+ 9m - 18 = 0`

factor out 9:

`\displaystyle m({m}^{} - 2 )+ 9(m - 2)= 0`

group:

`\displaystyle (m - 2)(m + 9) = 0`

hence,

`\displaystyle m_(1) = 2 \quad \text{and} \quad \displaystyle m _(2) = - 9`

remember that,

when we deal with logarithm equation we should always check the roots

let's check the root 1:

`\displaystyle \log_(4)( {2}^(2) ) \stackrel{?}{=} \log_(4)(18 - 7.2)`

simplify square:

`\displaystyle \log_(4)( 4) \stackrel{?}{=} \log_(4)(18 - 7.2)`

simplify multiplication:

`\displaystyle \log_(4)( 4) \stackrel{?}{=} \log_(4)(18 - 14)`

simplify substraction:

`\displaystyle \log_(4)( 4) \stackrel{?}{=} \log_(4)(4)`

simplify logarithm:

`\displaystyle 1 \stackrel{ \checkmark}{=} 1`

let's check root 2:

`\displaystyle \rm \log_(4)( { - 9}^(2) ) \stackrel{?}{=} \log_(4)(18 - 7.( - 9))`

simplify square:

`\displaystyle \rm \log_(4)( 81 ) \stackrel{?}{=} \log_(4)(18 + 63)`

simplify addition:

`\displaystyle \rm \log_(4)( 81 ) \stackrel{ \checkmark}{=} \log_(4)(81)`

therefore,

`\displaystyle m_(1) = 2 \quad \text{and} \quad \displaystyle m _(2) = - 9`

Answer 2

• m = {-9, 2}

Explanation:

• log₄ (m²) = log₄ (18 - 7m)

18 - 7m > 0 ⇒ 7m < 18 ⇒ m < 18/7

• m² = 18 - 7m
• m² + 7m - 18 = 0
• m² + 2*3.5m + 12.25 = 30.25
• (m + 3.5)² = 5.5²
• m = - 3.5 ± 5.5

• m = 2
• m = -9