Question

Логарим х по основанию 4 = логарифм корня из 2 по основанию 0,5

Answer 1

Log(4)x=log(0,5)√2
В скобках указываю основание
log(2²)x=log(1/2)2^(1/2)
Очень советую посмотреть свойства логарифмов, чтобы понять, как и почему я переношу числа из степени основания в множитель логарифма и т.д.
1/2*log(2)x=log(2^(-1))2^1/2
1/2log(2)x=-1/2log(2)2
log(2)x=-log(2)2
x=2^(-log(2)2)
x=2^(log(2)1/2)
x=√2

Answer 2

Solving the Equation

To solve the equation log₄(x) = log₀.₅(√2), we need to find the value of x.

Let's break down the equation step by step:

1. The equation log₄(x) represents the logarithm of x with a base of 4. 2. The equation log₀.₅(√2) represents the logarithm of √2 with a base of 0.5.

To solve this equation, we can use the property of logarithms that states: logₐ(b) = log(c) / log(a), where a is the base, b is the argument, and c is the base of the logarithm we want to convert to.

In our case, we can rewrite the equation as follows:

log₄(x) = log₀.₅(√2) log(x) / log(4) = log(√2) / log(0.5)

Now, let's calculate the values of the logarithms using the natural logarithm (log base e) for simplicity:

log(x) / log(4) = log(√2) / log(0.5) (log(x) / log(e)) / (log(4) / log(e)) = (log(√2) / log(e)) / (log(0.5) / log(e)) logₑ(x) / logₑ(4) = logₑ(√2) / logₑ(0.5)

Using the change of base formula, we can rewrite the equation as:

logₑ(x) / logₑ(4) = logₑ(√2) / logₑ(0.5) logₑ(x) / logₑ(4) = logₑ(√2) / logₑ(2⁻¹)

Since the logarithm of a square root (√2) with a base of 2⁻¹ is equal to 0.5, we can simplify the equation further:

logₑ(x) / logₑ(4) = 0.5

Now, we can solve for x by multiplying both sides of the equation by logₑ(4):

logₑ(x) = 0.5 * logₑ(4)

To find the value of x, we need to evaluate the right side of the equation using a calculator or a mathematical software.