Respuesta :
Answer:
x = 2
Step-by-step explanation:
Natural Logarithm (ln) is the logarithm is the base = e, therefore the Natural Logarithm is also applicable to all Logarithm Rules.
For this question, we apply the Power Rule:
[tex]\boxed{log_a(b^c)=c\cdot log_a(b)}[/tex]
[tex]2\cdot ln(4x-1)=ln(23x+3)[/tex]
[tex]2\cdot log_e(4x-1)=log_e(23x+3)[/tex]
[tex]log_e(4x-1)^2=log_e(23x+3)[/tex]
[tex](4x-1)^2=23x+3[/tex]
[tex]16x^2-8x+1=23x+3[/tex]
[tex]16x^2-31x-2=0[/tex]
[tex](16x+1)(x-2)=0[/tex]
[tex]x=-\frac{1}{16} \ or\ 2[/tex]
For logarithm, we have to check the answer with the argument restriction: [tex]\boxed{argument\geq 0}[/tex]
(i) [tex]4x-1\geq 0[/tex]
[tex]4x\geq 1[/tex]
[tex]x\geq \frac{1}{4}[/tex]
(ii) [tex]23x+3\geq 0[/tex]
[tex]23x\geq -3[/tex]
[tex]x\geq -\frac{3}{23}[/tex]
By combining the (i) & (ii), we can conclude that [tex]\boxed{x\geq \frac{1}{4}}[/tex]
Therefore, the answer [tex]x=-\frac{1}{16}[/tex] does not meet the restriction, and the answer is 2.
*Note: we can skip converting the [tex]ln[/tex] into [tex]log_e[/tex]:
[tex]2\cdot ln(4x-1)=ln(4x-1)^2[/tex]
After that, just eliminate the [tex]ln[/tex] on each side.
Answer:
Solution is x = 2
Step-by-step explanation:
- Let's write down the equation as given
[tex]2 \ln (4x - 1) = \ln(23x + 3)\\\\[/tex] [1]
- We are asked to solve for x
- Use the fact that 2 ln(x) = ln(x²) to get
[tex]2 \ln(4x - 1) = ln(4x-1)^2 \\\\\text{Equation 1 becomes}\\\\ln(x - 4)^2 = ln(23x + 3)\\\mathrm{Apply\:log\:rule:\:\:If}\:\log _b\left(f\left(x\right)\right)=\log _b\left(g\left(x\right)\right)\:\mathrm{then}\:f\left(x\right)=g\left(x\right)\\\\(4x - 1)^2 = (23x + 3)\quad\dots[2]\\[/tex]
- Expand left side (4x - 1)² using the identity (a + b)² = a² + 2ab + b²
in (4x - 1)² a = 4x, b = -1 so we get
(4x - 1)² = (4x)² + 2(4x)(-1) + (-1)²
→ 16x² -8x + 1
- Equate this expression to the right side expression in Equation 2:
16x² -8x + 1 = 23x + 3 [3]
- Subtract 3 from both sides of equation [3]
16x² - 8x + 1 - 3 = 23x + 3 -3
16x² - 8x - 2 = 23x
- Subtract 23x from both sides:
16x² - 8x - 2 - 23x = 23x - 23x
16x² - 31x - 2 = 0
- This is a quadratic equation that can be solved by the quadratic equation calculator or by factoring.
- Using a quadratic equation calculator we get the two solutions as
x = 2 and x = -1/16
- Check which of these values will result in real values for the ln functions
- At x = 2 ln(4x - 1) = 2 ln(4 (2) - 1) = 2ln(7) which is a real number
At x = 2 right side (23x + 3) is also a real number
|So x = 2 is a solution since it results in a real number on both sides
- Using x = -1/16 we get the left side as
2 · ln(4 · (- 1/16) - 1) = 2 · ln(-1/4 - 1) = 2 ln(-5/4)
Logs of negative numbers are not defined so the above expression is not defined and hence
x = -1/16 is not a valid solution
So the only solution to the given equation is x = 2
