Solving Exponential Equation

[colinjava]

I'm too old school to use a base other than e or 10. Years ago calculators would just use base e or 10, if you wanted another base you'd use the change of base formula, which I guess is just what the newer calculators do when you pick a strange base number when taking a log.

[Maths_.]

Solve for k over the real numbers:
4^k = 5^(k + 3)
Take the natural logarithm of both sides and use the identity log(a^b) = b log(a):
2 log(2) k = log(5) (k + 3)
Expand out terms of the right hand side:
2 log(2) k = log(5) k + 3 log(5)
Subtract k log(5) from both sides:
(2 log(2) - log(5)) k = 3 log(5)
Divide both sides by 2 log(2) - log(5):
Answer: k = (3 log(5))/(2 log(2) - log(5))
k≈-21.637..

[anbarnasimbihon]

How should I know when to use log and when to not?

[wiggles]

The regular way gives 3ln(5)/ln(4/5), and it is easy to check that this is equivalent to your answer. 3ln(5) = ln(5^3), and there is a rule ln(x)/ln(b) = log_b(x), thus 3ln(5)/ln(4/5) = log_0.8(125).

[mrflibble]

Excellent work, really enjoy your methods!

[sans]

4^k=5^(k+3)
=125(5^k)=4^k
divide by 4^k on both sides
125((5/4)^k)=1
(5/4)^k=1/125
k=ln(1/125) / ln(5/4)

1/125=125^-1=(5^3)^-1=5^-3
ln(1/125)=ln(5^-3)=-3ln5
ln4=2ln2
=>ln(5/4)=ln5-2ln2

[mgmartin]

I used a slide rule: I set the cursor on 0.8 (4/5) on the LL02 scale, then found 125 on the LL3 scale, and read 21.6 on the C scale. Since 0.8 is less than 1, the exponent has to be negative, so the answer reads -21.6!

[snlosrg]

Answer is 5x +2y -16=0. Thank you sir 🙏

[junaidhasrat]

Well built arms, like an ancient sensei.

[ziyadb]

Taking the log of both sides seems easier to remember.

[robertfield]

I enjoy the information. But the chalk board sends chills done my spine 😅

[Demon-syuw]

Ngl, i would of done like two extra steps there, cus my brain too dum

[lastchance]

Hate it when I have to use a calculator.

[reversatire]

Good luck putting log base 4/5th into a calculator. Why not log base 10 or e both sides:
ln(4/5)^k=ln(125) or k*ln(4/5)=ln(125)
k=ln(125)/ln(4/5)= -21.64

[HARSHSHARMA-dboh]

Dont you think sir we have to use calculator in this solution

[ibrahimsualez]

Im kind of person who dont know nothing about naths never undastood so wheb i see someone doing it i always compliment the person, what you are not intelligent on it you wil never im always like 🤒

[afshinfarzaadi]

[ (x+1)^(-1) ÷ (5x+6)/(1-2x) ]^(-1) +5(x-1) = (38x-8)/7
x=?

[cheeyeung]

I’m happy and glad my calculus tutor had better handwriting