Intereting Posts

a general continued fraction satisfying $\frac{(i+\Theta\sqrt{z})^m}{(i-\Theta\sqrt{z})^m}=\frac{(ik+\sqrt{z})^{m+1}}{(ik-\sqrt{z})^{m+1}}$
Consistency of Peano axioms (Hilbert's second problem)?
Is a differentiable function always continuous?
numbers' pattern
Integral of special asin $\int{(a^2-x^2)\sin^{-1}\left(-1+\frac{b}{\sqrt{a^2-x^2}}\right)\ dx}$
Largest Triangular Number less than a Given Natural Number
Show that $T\to T^*$ is an isomorphism (Where T is a linear transform)
Does an elementary solution exist to $x^2+1=y^3$?
Can $\int|f_n|d\mu \to \int |f|d\mu$ but not $\int|f_n – f|d\mu \to 0$?
Point closest to a set four of lines in 3D
Linear Representations coming from Permutation Representations
What is the most extreme set 4 or 5 nontransitive n-sided dice?
Improper integration involving complex analytic arguments
Prove that $\left | \frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a} \right | < \frac{1}{8}.$
Example of two-dimensional non-abelian Lie algebra?

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 $x=568.2993$.

- Why dividing by zero still works
- Find all the values of $(1+i)^{(1-i)}$
- Prove if $n^2$ is even, then $n$ is even.
- A closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ is …
- An infinitely powered expression
- Prove that $m+\frac{4}{m^2}\geq3$ for every $m > 0$

But is there any way to solve it algebraically/algorithmically?

- Defining division by zero
- Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
- Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
- If $f(x) = \sin \log_e (\frac{\sqrt{4-x^2}}{1-x})$ then find the range of this function.
- How do you construct a function that is continuous over $(0,1)$ whose image is the entire real line?
- Solving the equation $\ln(x)=-x$
- On the commutative property of multiplication (domain of integers, possibly reals)
- Learning how to flip equations
- How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
- Proving that the number $\sqrt{7 + \sqrt{50}} + \sqrt{7 - 5\sqrt{2}}$ is rational

I have solved a question similar to this before. In general, you can have a solution of the equation

$$ a^x=bx+c $$

in terms of the Lambert W-function

$$ -\frac{1}{\ln(a)}W_k \left( -\frac{1}{b}\ln(a) {{\rm e}^{-{\frac {c\ln(a) }{b}}}} \right)-{\frac {c}{b}}

\,.$$

Substituting $ a=1.01 \,,b=\frac{1}{2}\,,c=\frac{3}{2}$ and considering the values $k=0$ and $k=-1$, we get the zeroes $$x_1= -1.020199952\,, x_2=568.2993002 \,. $$

Polynomials don’t play nice with exponentials, so no. If you work hard, you might find an answer in terms of the Lambert W function, but if I did I wouldn’t feel much more enlightened.

A standard root finding procedure (such as Newton’s method) should solve the problem for you. You might also be interested in the Lambert W function, which will give you a “closed form” solution, assuming you have access to that function of course.

- Overlapping Polynomials
- How to prove an identity (Trigonometry Angles–Pi/13)
- Analyzing whether there is always a prime between $n^2$ and $n^2+n$
- the constant in the asymptotics of $\sum_{1\le k \le n} \frac{\varphi(k)}{k^2}$
- How do you prove that $M(N)=O(N^{1/2+\epsilon})$ from the Riemann Hypothesis?
- Picturing the discrete metric?
- Parabolic subgroups of $\mathrm{Sl}_n$ are the ones that stabilize some flag
- One of the diagonals in a hexagon cuts of a triangle of area $\leq 1/6^{th}$ of the hexagon
- Finding Smith normal form of matrix over $ \mathbb{R} $
- Can we think of a chain homotopy as a homotopy?
- Is it true that $f(x,y)=\frac{x^2+y^2}{xy-t}$ has only finitely many distinct positive integer values with $x$, $y$ positive integers?
- How to find $\lim _{ n\to \infty } \frac { ({ n!) }^{ 1\over n } }{ n } $?
- Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?
- dice problem – throws necessary for sum multiple of n
- Poincare dual of unit circle