Solve: $x^3=0.5625$

Question

Solve: $x^3=0.5625$

$$x^3=0.5625/\log_3$$ $$\log_3x^3=\log_3 0.5625$$ $$3\log_3 x=\log_3 0.5625$$ $$\log_3 x=\frac{\log_3 0.5625}{3}$$

How to evaluate $\log_3 0.5625$?

How to change basis in logarithms for simplicity?

For example, if we have $\log_3 0.5625$, how to convert this logarithm such that the basis is not $3$, but $10$ (on my calculator, I only have $10$ as a basis)?

Why exactly are you trying to involve logarithms in this? All you need to do is take the cube root. — Sebastian Schoennenbeck, Jun 06 '17 at 09:59
Why do you want to use base $3$? Base $10$ works as well, but $\sqrt[3]{0.5625}=0.82548181222365667096865248810227$ would be easier. — , Jun 06 '17 at 10:01
Why not simply $;x=\sqrt[3]{0.5625};$ ? You're going to use a calculator, anyway: what in the world do you need to mess up with logarithms for? — DonAntonio, Jun 06 '17 at 10:01
@Sebastian Schoennenbeck, because we can't calculate third root directly. We can calculate second and fourth root, and than aproximate third root. — user7920621, Jun 06 '17 at 10:05
@DonAntonio, Yes but my calculator can calculate only even roots (2,4,...). — user7920621, Jun 06 '17 at 10:13
@user7920621 Can't you do $\sqrt[3]{0.5625}=0.5625^{1/3}$ with your calculator? What calculator model do you have? — projectilemotion, Jun 06 '17 at 10:22

NDewolf · Answer 1 · 2017-06-06T10:06:57.803

2

You can use the formula $$\log_a(x) =\frac{\log_b(x)}{\log_b(a)}$$ where $a, b$ are positive numbers. In your case these are 3 and 10.

edited Jun 06 '17 at 10:06

answered Jun 06 '17 at 10:00

NDewolf

1,663

What is the base for logarithm in numerator? – user7920621 Jun 06 '17 at 10:02
You can choose that freely. – Jun 06 '17 at 10:04
Yes i should have clarified. My apologies. – NDewolf Jun 06 '17 at 10:05
So, we can freely choose $a,b$? – user7920621 Jun 06 '17 at 10:16
Yes, as long as they are positive and not equal to 1. – NDewolf Jun 06 '17 at 10:16
@NDewolf, I get $\log_3 x=\frac{\log_{10}x}{\log_{10}3}=\frac{\log_{10}x}{0.48}$. From here, how to evaluate $x$? – user7920621 Jun 06 '17 at 10:27
That expression is equal to the right hand side of the formula in your question, which you can calculate in the same way. Then you can isolate the $\log_{10}(x)$ factor and invert the formula to obtain $x$. – NDewolf Jun 06 '17 at 10:29

Solve: $x^3=0.5625$

1 Answers1