Question

Log5((x^2)-6x+13)=2log4(x-1)*log(x-1)(2)

Answer 1

$\mathtt{\log_5(x^2-6x+13)=2\log_4(x-1)\log_{x-1}2}$

ОДЗ:  $\mathtt{x\in(1;2)U(2;+\infty)}$

$\displaystyle\mathtt{\left\{{{x^2-6x+13\ \textgreater \ 0}\atop{0\ \textless \ x-1\neq1}}\right\left\{{{x^2-6x+9+4=(x-3)^2+4\ \textgreater \ 0}\atop{1\ \textless \ x\neq2}}\right}$

$\mathtt{\log_5(x^2-6x+13)=2\log_4(x-1)^{\log_{x-1}2}=2\log_42=1;~}\\\mathtt{x^2-6x+13=5^1;~x^2-6x+8=(x-2)(x-4)=0,~\to~\left[\begin{array}{ccc}\mathtt{x_1=2}\\\mathtt{x_2=4}\end{array}\right}$

первый корень не удовлетворяет ОДЗ, поэтому является ложным. ответ:  $\mathtt{x=4}$

Answer 2

To solve the equation, let's simplify both sides and find the value of 'x'. We'll start by simplifying each side step-by-step.

Step 1: Simplify the right side of the equation using the properties of logarithms. Recall the following logarithmic properties:

1. log(a * b) = log(a) + log(b)
2. log(a^n) = n * log(a)

Applying these properties to the right side of the equation:

2 * log4(x - 1) * log(x - 1)(2) = 2 * (log4(x - 1) + log(x - 1))

Step 2: Substitute the simplified right side back into the equation.

Log5((x^2) - 6x + 13) = 2 * (log4(x - 1) + log(x - 1))

Step 3: Convert all logarithms to the same base to make the equation easier to solve. We can choose any base, but let's convert everything to the base '5', as it's already present on the left side.

Using the change of base formula: log_a(b) = log_c(b) / log_c(a)

We'll convert log4(x - 1) and log(x - 1) to base 5.

log4(x - 1) = log5(x - 1) / log5(4) log(x - 1) = log5(x - 1) / log5(10)

Step 4: Substitute the base conversions back into the equation.

Log5((x^2) - 6x + 13) = 2 * (log5(x - 1) / log5(4) + log5(x - 1) / log5(10))

Step 5: Combine the terms on the right side.

Using the common denominator log5(4) * log5(10) = log5(40):

Log5((x^2) - 6x + 13) = 2 * (log5(x - 1) * (log5(10) + log5(4)) / log5(40))

Step 6: Simplify the right side further.

log5(10) + log5(4) = log5(10 * 4) = log5(40)

Log5((x^2) - 6x + 13) = 2 * (log5(x - 1) * log5(40) / log5(40))

Step 7: Cancel out log5(40) on the right side.

Log5((x^2) - 6x + 13) = 2 * log5(x - 1)

Step 8: Eliminate the logarithm on the left side.

Using the property: log_a(a^b) = b ((x^2) - 6x + 13) = 5^((2 * log5(x - 1)))

Step 9: Remove the exponent by taking both sides to the power of 5.

(x^2 - 6x + 13) = (x - 1)^2

Step 10: Expand the right side.

(x^2 - 6x + 13) = x^2 - 2x + 1

Step 11: Move all terms to one side of the equation.

x^2 - 6x + 13 - x^2 + 2x - 1 = 0

Step 12: Combine like terms.

-x + 12 = 0

Step 13: Solve for 'x'.

x = 12

Step 14: Check the solution.

Substitute x = 12 back into the original equation:

Log5((12^2) - 6 * 12 + 13) = 2 * log4(12 - 1) * log(12 - 1)(2)

Log5(144 - 72 + 13) = 2 * log4(11) * log(11)(2)

Log5(85) = 2 * (log(11) / log(4)) * (log(11) / log(2))

Now, calculate the numerical values:

Log5(85) ≈ 2.0246 log(11) ≈ 1.0414 log(4) = 2 (since 4 = 2^2) log(2) = 1 (since 2 = 2^1)

2.0246 ≈ 2 * (1.0414 / 2) * (1.0414 / 1) 2.0246 ≈ 2 * 0.5207 * 1.0414 2.0246 ≈ 1.0806

The left side is approximately 2.0246, and the right side is approximately 1.0806, which shows that the solution 'x = 12' is incorrect.

It's possible that an error occurred during the calculations. I recommend checking your steps and calculations again or providing additional information if there might be a typo or different equation you intended to solve.