All Questions
Tagged with algebra-precalculus logarithms
84
questions
10
votes
3
answers
2k
views
Solving the equation $3 + x = 2 (1.01^x)$ for $x$
This equation clearly cannot be solved using logarithms.
$$3 + x = 2 (1.01^x)$$
Now it can be solved using a graphing calculator or a computer and the answer is $x = -1.0202$ and $x=568.2993$.
But ...
- 111
18
votes
4
answers
165k
views
How to figure out the log of a number without a calculator?
I have seen people look at log (several digit number) and rattle off the first couple of digits.
I can get the value for small values (aka the popular or easy to know roots), but is there a formula. ...
- 7,362
6
votes
4
answers
996
views
What is the solution to the equation $9^x - 6^x - 2\cdot 4^x = 0 $?
I want to solve: $$9^x - 6^x - 2\cdot 4^x = 0 $$
I was able to get to the equation below by substituting $a$ for $3^x$ and $b$ for $2^x$:
$$ a^2 - ab - 2b^2 = 0 $$
And then I tried
\begin{align}x ...
- 219
9
votes
5
answers
7k
views
Solve $2^x=x^2$
I've been asked to solve this and I've tried a few things but I have trouble eliminating $x$. I first tried taking the natural log:
$$x\ln \left( 2\right) =2\ln \left( x\right)$$
$$\dfrac {\ln \left( ...
- 7,177
5
votes
3
answers
3k
views
Prove that $n \ln(n) - n \le \ln(n!)$ without Stirling
I need to prove that $n \ln(n) - n \le \ln(n!)$. I have solved this but I've used the Stirling substitution for the factorial term which does not seem good to me in this proof. I am sure that there ...
- 3,709
8
votes
4
answers
2k
views
How $a^{\log_b x} = x^{\log_b a}$?
This actually triggered me in my mind from here. After some playing around I notice that the relation $a^{\log_b x} = x^{\log_b a}$ is true for any valid value of $a,b$ and $x$. I am very inquisitive ...
- 22.5k
16
votes
9
answers
4k
views
Intuition behind logarithm change of base
I try to understand the actual intuition behind the logarithm properties and came across a post on this site that explains the multiplication and thereby also the division properties very nicely:
...
- 2,493
5
votes
3
answers
1k
views
Why is $x^{\log_x n}=n$?
I'm currently doing a couple of exercises on logarithmic expressions, and I was a bit confused when presented with the following: $5^{\log_5 17}$.
The answer is $17$, which is the argument of the ...
- 1,005
6
votes
3
answers
560
views
If $\log(x^2+2ax)=\log(4x-4a-13)$ has only one solution, find the exhaustive set of values of $a$
If $\log(x^2+2ax)=\log(4x-4a-13)$ has only one solution, then what is the exhaustive set of values of $a$ ?
This question was asked at Equation $\log(x^2+2ax)=\log(4x-4a-13)$ has only one solution; ...
- 3,835
5
votes
3
answers
9k
views
Write the expression $\log(\frac{x^3}{10y})$ in terms of $\log x$ and $\log y$
What is the answer for this? Write the expression in terms of $\log x$ and $\log y$ $$\log\left(\dfrac{x^3}{10y}\right)$$
This is what I got out of the equation so far. the alternate form assuming $x$...
- 781
3
votes
2
answers
260
views
Find $n$ satisfying the equation $[\log_21]+[\log_22]+[\log_23]+\dots[\log_2n]=1538 $
If $[\cdot]$ denotes greatest integer function, then what is the value of natural number $n$ satisfying the equation $$[\log_21]+[\log_22]+[\log_23]+\dots[\log_2n]=1538 ?$$
My try:
Note that
$$0+1\...
- 6,590
16
votes
1
answer
1k
views
Are Base Ten Logarithms Relics?
Just interested in your thoughts regarding the contention that
the pre-eminence of base ten logarithms is a relic from
pre-calculator days.
Firstly I understand that finding the (base-10) ...
- 7,789
15
votes
7
answers
2k
views
Given $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}$ show that $x^{b+c-a}\cdot y^{c+a-b}\cdot z^{a+b-c} = 1$
Given:
$$\dfrac{\log x}{b-c}=\dfrac{\log y}{c-a}=\dfrac{\log z}{a-b}$$
We have to show that :
$$x^{b+c-a}\cdot y^{c+a-b}\cdot z^{a+b-c} = 1$$
I made three equations using cross multiplication :
$$...
- 3,517
14
votes
6
answers
9k
views
Is "ln" (natural log) and "log" the same thing if used in this answer?
Find $x$ for $4^{x-4} = 7$.
Answer I got, using log, was ${\log(7)\over 2\log(2)} + 4$
but the actual answer was ${\ln(7)\over2\ln(2)} + 4$
I plugged both in my calculator and turns out both are ...
- 444
13
votes
1
answer
9k
views
Proof of concavity of log function
Does anybody have a proof of the concavity of the $\log{x}$ that does not use calculus?
- 139