Question

7^log(49) (x+1) = 5^log(125) (3x-1)

Answer 1

ОТВЕТ
3

см скриншот
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Answer 2

To solve the equation 7^log(49)(x+1) = 5^log(125)(3x-1), we'll first simplify both sides using the properties of logarithms and then solve for x. Let's begin:

Step 1: Simplify both sides using the properties of logarithms.

First, we know that log(a^b) = b*log(a). So, we can rewrite the equation as:

log(49)(x+1) = log(125)(3x-1)

Step 2: Now, let's use the logarithmic identity that log(a*b) = log(a) + log(b). We'll apply this to both sides of the equation:

log(49) + log(x+1) = log(125) + log(3x-1)

Step 3: We need to find the values of log(49), log(125), log(x+1), and log(3x-1). For simplicity, let's use the base 10 logarithm:

log(49) = log(7^2) = 2*log(7)

log(125) = log(5^3) = 3*log(5)

Step 4: Now, let's rewrite the equation with the substituted values:

2log(7) + log(x+1) = 3log(5) + log(3x-1)

Step 5: Move all terms containing x to one side of the equation:

log(x+1) - log(3x-1) = 3log(5) - 2log(7)

Step 6: Apply the logarithmic identity log(a) - log(b) = log(a/b):

log[(x+1)/(3x-1)] = 3log(5) - 2log(7)

Step 7: Now, let's eliminate the logarithm by using the exponential function. Since the base of the logarithm is not specified, we'll use the base 10 exponential:

(x+1)/(3x-1) = 10^(3log(5) - 2log(7))

Step 8: Simplify the right-hand side:

10^(3log(5) - 2log(7)) = 10^(log(5^3) - log(7^2)) = 10^(log(125/49)) = 125/49

Step 9: Now, we have a simple algebraic equation:

(x+1)/(3x-1) = 125/49

Step 10: Cross-multiply and solve for x:

49(x+1) = 125(3x-1)

49x + 49 = 375x - 125

49 + 125 = 375x - 49x

174 = 326x

x = 174/326

x ≈ 0.5331

So, the approximate value of x that satisfies the given equation is 0.5331.