Mathematics: Write in logarithmic form:

Step-by-step explanation:The given function isTo find the inverse of this function, we have to replace for , and then isolate , as followsThen, we have to switch variables, as followsTherefore, the inverse of the given function is The graph attached shows both functions. The blue line is the inverse of...

Write in logarithmic form: - id9605619, MGA20078

Ответ:

mrsali35

12.02.2022

Mathematics

Left-hand side: $\displaystyle -\frac{1}{2}\, \ln(5)$ .

Right-hand side: $\displaystyle \frac{1}{2}\, \ln(5) - \ln(5)$ .

Step-by-step explanation:

Apply the logarithm power rule: $\ln \left(x^{a}\right) = a\, \ln (x)$ for all x 0 .

This property is not only true for logarithm to the base , but for other bases, as well.

Take the logarithm (to the base ) of the left-hand side of this equation:

$\displaystyle \ln \left(5^{-1/2}\right) = (-1/2)\, \ln(5)$ .

For the right-hand side of this equation, consider the logarithm quotient rule:

$\displaystyle \ln \left(\frac{a}{b}\right) = \ln(a) - \ln (b)$ for all a 0 and b 0 .

Indeed, on the right-hand side of this equation, $\sqrt{5} 0$ and 5 0 . Therefore:

$\displaystyle \ln\left(\frac{\sqrt{5}}{5}\right) = \ln\left(\sqrt{5}\right) - \ln(5)$ .

This expression could be further simplified. Notice that $\sqrt{x}$ is equivalent to $x^{1/2}$ for all $x \ge 0$ . (Think about how $\sqrt{x} \cdot \sqrt{x} =x$ whereas $x^{1/2} \cdot x^{1/2} = x^{(1/2) + (1/2)} = x$ .)

Therefore, $\ln \left(\sqrt{5}\right)$ would be equivalent to $\ln\left(5^{1/2}\right)$ . Apply the logarithm power rule to show that $\displaystyle \ln\left(5^{1/2}\right) = \frac{1}{2}\, \ln(5)$ .

$\begin{aligned} \text{R.H.S.} &= \ln\left(\frac{\sqrt{5}}{5}\right) \\ &= \ln\left(\sqrt{5}\right) - \ln(5) \\ &= \frac{1}{2}\, \ln(5) - \ln (5) = -\frac{1}{2}\, \ln(5)\end{aligned}$ .