Math virtual assistant

Calculators Topics Go Premium About Snapxam
ENGESP
Topics

Base change formula of logarithms Calculator

Get detailed solutions to your math problems with our Base change formula of logarithms step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here!

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

1

Solved example of base change formula of logarithms

$\lim_{x\to0}\left(\frac{5^x-1}{\ln\left(1+x\right)}\right)$
2

As the limit results in indeterminate form, we can apply L'Hôpital's rule

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(5^x-1\right)}{\frac{d}{dx}\left(\ln\left(1+x\right)\right)}\right)$
3

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\lim_{x\to0}\left(\frac{d}{dx}\left(5^x-1\right)\frac{1+x}{\frac{d}{dx}\left(1+x\right)}\right)$
4

The derivative of a sum of two functions is the sum of the derivatives of each function

$\lim_{x\to0}\left(\left(\frac{d}{dx}\left(5^x\right)+\frac{d}{dx}\left(-1\right)\right)\frac{1+x}{\frac{d}{dx}\left(1+x\right)}\right)$
5

The derivative of the constant function ($-1$) is equal to zero

$\lim_{x\to0}\left(\frac{d}{dx}\left(5^x\right)\frac{1+x}{\frac{d}{dx}\left(1+x\right)}\right)$
6

Applying the derivative of the exponential function

$\lim_{x\to0}\left(5^x\ln\left(5\right)\frac{d}{dx}\left(x\right)\frac{1+x}{\frac{d}{dx}\left(1+x\right)}\right)$
7

The derivative of the linear function is equal to $1$

$\lim_{x\to0}\left(1.60945^x\frac{1+x}{\frac{d}{dx}\left(1+x\right)}\right)$
8

The derivative of a sum of two functions is the sum of the derivatives of each function

$\lim_{x\to0}\left(1.60945^x\frac{1+x}{\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(x\right)}\right)$
9

The derivative of the constant function ($1$) is equal to zero

$\lim_{x\to0}\left(1.60945^x\frac{1+x}{\frac{d}{dx}\left(x\right)}\right)$
10

The derivative of the linear function is equal to $1$

$\lim_{x\to0}\left(1.60945^x\left(1+x\right)\right)$
11

Solve the product $1.60945^x\left(1+x\right)$

$\lim_{x\to0}\left(\left(1.6094+1.6094x\right)5^x\right)$
12

Multiplying polynomials $5^x$ and $1.6094+1.6094x$

$\lim_{x\to0}\left(1.60945^x+1.6094x5^x\right)$
13

The limit of a sum of two functions is equal to the sum of the limits of each function: $\displaystyle\lim_{x\to c}(f(x)\pm g(x))=\lim_{x\to c}(f(x))\pm\lim_{x\to c}(g(x))$

$\lim_{x\to0}\left(1.60945^x\right)+\lim_{x\to0}\left(1.6094x5^x\right)$
14

Evaluating the limit when $x$ tends to $0$

$\left(5^0\right)\left(1.6094\right)+\lim_{x\to0}\left(1.6094x5^x\right)$

Calculate the power $5^0$

$\left(1\right)\left(1.6094\right)+\lim_{x\to0}\left(1.6094x5^x\right)$

Any expression multiplied by $1$ is equal to itself

$1.6094+\lim_{x\to0}\left(1.6094x5^x\right)$
15

Simplifying

$1.6094+\lim_{x\to0}\left(1.6094x5^x\right)$
16

Evaluating the limit when $x$ tends to $0$

$1.6094+\left(5^0\right)\left(1.6094\right)\left(0\right)$

Any expression multiplied by $0$ is equal to $0$

$1.6094+0$
17

Simplifying

$1.6094+0$
18

Add the values $1.6094$ and $0$

$1.6094$

Answer

$1.6094$

Struggling with math?

Access detailed step by step solutions to millions of problems, growing every day!