Question

Log(2) x + log(8)x =8

Answer 1

$log_2x+log_8x=8; \\ log_8x= \frac{log_2x}{log_28}= \frac{log_2x}{log_2{2^3}}= \frac{1}{3}log_2x; \\ log_2x+\frac{1}{3}log_2x=8 ; \quad \frac{4}{3}log_2x=8; \quad log_2x=6 \rightarrow x=2^6=64.$
Ответ: уравнение имеет единственный корень x=64

Answer 2

To solve the equation log(2)x + log(8)x = 8, we can use the properties of logarithms to simplify it.

First, let's apply the property that states log(a) + log(b) = log(ab):

log(2)x + log(8)x = log(2x) + log(8x)

Next, we can use another property that says log(a^n) = n * log(a):

log(2x) + log(8x) = log(2 * x) + log(8 * x) = log(16 * x^2)

Now, we have the equation log(16 * x^2) = 8.

Since log(16 * x^2) = 8 is a logarithmic equation, we can rewrite it in exponential form as:

16 * x^2 = 10^8

To solve for x, we need to isolate x^2, so divide both sides of the equation by 16:

x^2 = 10^8 / 16

x^2 = 6250000

Taking the square root of both sides, we get:

x = ±√6250000

Therefore, x = ±2500.

So the two solutions to the equation log(2)x + log(8)x = 8 are x = 2500 and x = -2500.