List of top Mathematics Questions on Exponential and Logarithmic Functions

If for all \(x\in\mathbb{R}\), \[ \frac{1}{3} < \frac{x^2-2x+4}{x^2+2x+4} <3, \] then the values of \[ \frac{9\cdot 3^{2x}+6\cdot 3^x+4} {9\cdot 3^{2x}-6\cdot 3^x+4} \] lie between

Let $f(x) = \log_e(9x)$ for $x > 0$ and $h(x) = f(x) + f(x^2) + f(x^3)$. Then the value of $h(\frac{1}{3}e^{1/3})$ is equal to

If $y=e^{2x}+sin~x$, then $2y^{\prime\prime}-5y^{\prime}+2y=$}

The number of real roots of the equation \[ e^{3x}-2e^{2x}-e^{x}+2=0 \] is

If \[ \log_2(x-1)+\log_2(x-3)=3, \] then the value of \(x\) is

If \[ \log_a 2+\log_a 5=1, \] then the value of \(a\) is:

If \[ \log_2(x-1)+\log_2(x+1)=3, \] then \(x\) is equal to:

Solve for \(x\): \[ x+\log_{15}(5+3^x)=x\log_{15}5+\log_{15}24 \]

Solve for \(x\): \[ x+\log_{15}(5+3^x)=x\log_{15}5+\log_{15}24 \]

The range of the function $f(x)=\left(\frac{1}{3}\right)^{3+\sin x}$ is

If \(\log_8 x = \frac{1}{3}\), find the value of \(x\).

If $\log_a b = 2$, then $b = ?$

Find the function \( f \) which satisfies the equation \( \frac{df}{dx} = 2f \), given that \( f(0) = e^3 \)

Evaluate \[ \int \frac{1}{1 + x^2} \, dx = ? \]

Evaluate the integral: \[ \int \frac{\sec^2 x}{(\sec x+\tan x)^{5/2}}dx \]

\( \int \frac{x}{(1-x^2)\sqrt{2 - x^2} dx = \)}

If $f : \mathbb{R}^+ \to \mathbb{R}$ is defined as $f(x) = \log_a x$ where $a>0$ and $a \neq 1$, prove that $f$ is a bijection. (R$^+$ is the set of all positive real numbers.)