Mathematics, 04.07.2021 01:00, MrKrinkle77

Solve the logarithmic equation

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

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answered: 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.

answered: 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 .

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

answered: Guest

Not enough information

answered: Guest

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

the function given is y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 = -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24