Mathematics, 04.07.2021 01:00 MrKrinkle77

Solve the logarithmic equation

$log_{6}(2x - 6) + log_{6}x = 2$

Answers: 1

Show answers

Answers

Answer from: CourtGard

x=6

Step-by-step explanation:

We want to solve the equation:

$\displaystyle \log_6(2x-6)+\log_6x=2$

Recall the property:

$\displaystyle \log_bx+\log_by=\log_b(xy)$

Hence:

$\log_6(x(2x-6))=2$

Next, recall that by the definition of logarithms:

$\displaystyle \log_b(a)=c\text{ if and only if } b^c=a$

Therefore:

6^2=x(2x-6)

Solve for x. Simplify and distribute:

36=2x^2-6x

We can divide both sides by two:

x^2-3x=18

Subtract 18 from both sides:

x^2-3x-18=0

Factor:

(x-6)(x+3)=0

Zero Product Property:

$x-6=0\text{ or } x+3=0$

Solve for each case. Hence:

$x=6\text{ or } x=-3$

Next, we must check the solutions for extraneous solutions. To do so, we can simply substitute the solutions back into the original equations and examine its validity.

Checking x = 6:

$\displaystyle \begin{aligned} \log_{6}(2(6)-6)+\log_{6}6&\stackrel{?}{=} 2 \\ \\ \log_6(12-6)+(1)&\stackrel{?}{=}2 \\ \\ \log_6(6)+1&\stackrel{?}{=}2 \\ \\ 1+1=2&\stackrel{\checkmark}{=}2\end{aligned}$

Hence, x = 6 is indeed a solution.

Checking x = -3:

$\displaystyle\begin{aligned} \log_6(2(-3)-6) + \underbrace{\log_6-3}_{\text{und.}} &\stackrel{?}{=} 2\\ \\ \end{aligned}$

Since the second term is undefined, x = -3 is not a solution.

Therefore, our only solution is x = 6.

Answer from: Animallover100

x = 6

Step-by-step explanation:

The given logarithmic equation is ,

$\implies log_{6}(2x - 6) + log_{6}x = 2$

We can notice that the bases of both logarithm is same . So we can use a property of log as ,

$\bf \to log_a b + log_a c = log_a {( ac)}$

So we can simplify the LHS and write it as ,

$\implies log_{6} \{ x ( 2x - 6 )\} = 2$

Now simplify out x(2x - 6 ) . We get ,

$\implies log_6 ( 2x^2 - 6x ) = 2$

Again , we know that ,

$\bf \to log_a b = c , a^c = b$

Using this we have ,

$\implies 2x^2 - 6x = 6^2 \\\\\implies 2x^2 - 6x -36 = 0$

Now simplify the quadratic equation ,

$\implies x^2 - 3x - 18 = 0 \\\\\implies x^2 -6x + 3x -18=0\\\\\implies x( x -6) +3( x - 6 ) = 0 \\\\\implies (x-6)(x+3) = 0 \\\\\implies x = 6 , -3$

Since logarithms are not defined for negative numbers or zero , therefore ,

$\implies 2x - 6 0 \\\\\implies x 3$

Therefore the equation is not defined at x = -3 . Hence the possible value of x is 6 .

$\implies \underline{\underline{ x \quad = \quad 6 }}$

Answer from: Quest

the answer is 844

step-by-step explanation:

it is 844 because if you add 836.50 + 7.5 you will get 844

Answer from: Quest

"it's not the heat, it's the humidity". that's a partly valid phrase you may have heard in the summer, but it's actually both. the heat index, also known as the apparent temperature, is what the temperature feels like to the human body when relative humidity is combined with the air temperature. this has important considerations for the human body's comfort. when the body gets too hot, it begins to perspire or sweat to cool itself off. if the perspiration is not able to evaporate, the body cannot regulate its temperature. evaporation is a cooling process. when perspiration is evaporated off the body, it effectively reduces the body's temperature. when the atmospheric moisture content (i.e. relative humidity) is high, the rate of perspiration from the body decreases. in other words, the human body feels warmer in humid conditions. the opposite is true when the relative humidity decreases because the rate of perspiration increases. the body actually feels cooler in arid conditions. there is direct relationship between the air temperature and relative humidity and the heat index, meaning as the air temperature and relative humidity increase (decrease), the heat index increases (decreases).

step-by-step explanation: