Kye
2021-08-15
Answered

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hajavaF

Answered 2021-08-16
Author has **90** answers

We are given:
$5({10}^{x}-6)=7$

Divide both sides by 5:${10}^{x}-6=1.4$

Take the (common) logarithm of both sides:${\mathrm{log}10}^{x}-6=\mathrm{log}1.4$

Recall that$\mathrm{log}b\cdot {b}^{x}=x$ and since the common logarithm is base 10, we write:
$x-6=\mathrm{log}1.4$

Add 6 to both sides:$x=6+\mathrm{log}1.4\sim 6.146$

Divide both sides by 5:

Take the (common) logarithm of both sides:

Recall that

Add 6 to both sides:

asked 2021-12-30

Prove that:

$\underset{x\to 0}{lim}\frac{\mathrm{ln}\left(\mathrm{cos}x\right)}{\mathrm{ln}(1-\frac{{x}^{2}}{2})}=1$

without LHopitals rule.

without LHopitals rule.

asked 2022-03-15

Arithmetically showing that $\frac{\mathrm{log}(x+1)}{\mathrm{log}\left(x\right)}<\frac{x+1}{x}$

Is there a possibility that this can be shown arithmetically? By arithmetically, I mean not looking at the graph.

$\frac{\mathrm{log}(x+1)}{\mathrm{log}\left(x\right)}<\frac{x+1}{x}$

Thank You

Is there a possibility that this can be shown arithmetically? By arithmetically, I mean not looking at the graph.

Thank You

asked 2022-03-26

Could you describe this function as "logarithmic"?

$f\left(x\right)=\frac{1}{\sqrt{x}}$

As x increases, the value of$f\left(x\right)$ decreases, but the decrease tapers off quickly as $x$ gets larger, and if you plot the graph of $f\left(x\right)$ , the shape looks kind of like an upside-down logarithm. Would it be correct to describe this function as declining logarithmically, as $x$ increases?

As x increases, the value of

asked 2022-05-15

How do I evaluate this logarithm expression?

I think the answer for this is 0.01, but I'm not sure. Could someone explain the steps in solving the following for $(x/y)$:

$10{\mathrm{log}}_{10}(x/y)=-20$

I've tried putting $\frac{-20}{10{\mathrm{log}}_{10}}$ in Wolfram Alpha, but the answer doesn't look like what I was expecting.

I think the answer for this is 0.01, but I'm not sure. Could someone explain the steps in solving the following for $(x/y)$:

$10{\mathrm{log}}_{10}(x/y)=-20$

I've tried putting $\frac{-20}{10{\mathrm{log}}_{10}}$ in Wolfram Alpha, but the answer doesn't look like what I was expecting.

asked 2021-10-29

The One-to-One Property of natural logarithms states that if ln x = ln y, then ________.

asked 2022-04-05

Equation $\mathrm{log}({x}^{2}+2ax)=\mathrm{log}(4x-4a-13)$ has only one solution; then exhaustive set of values of $a$ is

asked 2022-05-19

Are there other cases similar to Herglotz's integral ${\int}_{0}^{1}\frac{\mathrm{ln}(1+{t}^{4+\sqrt{15}})}{1+t}\text{}\mathrm{d}t$?

This post of Boris Bukh mentions amazing Gustav Herglotz's integral

${\int}_{0}^{1}\frac{\mathrm{ln}(1+{t}^{\phantom{\rule{thinmathspace}{0ex}}4\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\sqrt{\phantom{A}\phantom{\rule{thinmathspace}{0ex}}15\phantom{\rule{thinmathspace}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}})}{1+t}\text{}\mathrm{d}t=-\frac{{\pi}^{2}}{12}(\sqrt{15}-2)+\mathrm{ln}2\cdot \mathrm{ln}(\sqrt{3}+\sqrt{5})+\mathrm{ln}\frac{1+\sqrt{5}}{2}\cdot \mathrm{ln}(2+\sqrt{3}).$

I wonder if there are other irrational real algebraic exponents α such that the integral

${\int}_{0}^{1}\frac{\mathrm{ln}(1+{t}^{\phantom{\rule{thinmathspace}{0ex}}\alpha})}{1+t}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t$

has a closed-form representation? Is there a general formula giving results for such cases?

Are there such algebraic $\alpha $ of degree $>2$?

This post of Boris Bukh mentions amazing Gustav Herglotz's integral

${\int}_{0}^{1}\frac{\mathrm{ln}(1+{t}^{\phantom{\rule{thinmathspace}{0ex}}4\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\sqrt{\phantom{A}\phantom{\rule{thinmathspace}{0ex}}15\phantom{\rule{thinmathspace}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}})}{1+t}\text{}\mathrm{d}t=-\frac{{\pi}^{2}}{12}(\sqrt{15}-2)+\mathrm{ln}2\cdot \mathrm{ln}(\sqrt{3}+\sqrt{5})+\mathrm{ln}\frac{1+\sqrt{5}}{2}\cdot \mathrm{ln}(2+\sqrt{3}).$

I wonder if there are other irrational real algebraic exponents α such that the integral

${\int}_{0}^{1}\frac{\mathrm{ln}(1+{t}^{\phantom{\rule{thinmathspace}{0ex}}\alpha})}{1+t}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t$

has a closed-form representation? Is there a general formula giving results for such cases?

Are there such algebraic $\alpha $ of degree $>2$?