Question

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

Answer 1

Answer:

• m = {-9, 2}

Step-by-step 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

Answer 2

Answer:

$\displaystyle m_{1} = 2 \quad \text{and} \quad \displaystyle m _{2} = - 9$

Step-by-step 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,

$\displaystyle m_{1} = 2 \quad \text{and} \quad \displaystyle m _{2} = - 9$