Convert Exponentials and Logarithms
In convert Exponentials and Logarithms we will mainly discuss how to change the logarithm expression to Exponential expression and conversely from Exponential expression to logarithm expression.
To discus about convert Exponentials and Logarithms we need to first recall about logarithm and exponents.
The logarithm of any number to a given base is the index of the power to which the base must be raised in order to equal the given number. Thus, if aˣ = N, x is called the logarithm of N to the base a.
For example:
1. Since 3⁴ = 81, the logarithm of 81 to base 3 is 4.
2. Since 10¹ = 10, 10² = 100, 10³ = 1000, ………….
The natural number 1, 2, 3, …… are respectively the logarithms of 10, 100, 1000, …… to base 10.
The logarithm of N to base a is usually written as log₀ N, so that the same meaning is expressed by the two equations
a^{x} = N; x = log_{a} N
Examples on convert Exponentials and Logarithms
1. Convert the following exponential form to logarithmic form:
(i) 10
^{4} = 10000
Solution:
10
^{4} = 10000
⇒ log
_{10} 10000 = 4
(ii) 3
^{5} = x
Solution:
3
^{5} = x
⇒ log
_{3} x = 5
(iii) (0.3)
^{3} = 0.027
Solution:
(0.3)
^{3} = 0.027
⇒ log
_{0.3} 0.027 = 3
2. Convert the following logarithmic form to exponential form:
(i) log
_{3} 81 = 4
Solution:
log
_{3} 81 = 4
⇒ 3
^{4} = 81, which is the required exponential form.
(ii) log
_{8} 32 = 5/3
Solution:
log
_{8} 32 = 5/3
⇒ 8
^{5/3} = 32
(iii) log
_{10} 0.1 = 1
Solution:
log
_{10} 0.1 = 1
⇒ 10
^{1} = 0.1.
3. By converting to exponential form, find the values of following:
(i) log
_{2} 16
Solution:
Let log
_{2} 16 = x
⇒ 2
^{x} = 16
⇒ 2
^{x} = 2
^{4}
⇒ x = 4,
Therefore, log
_{2} 16 = 4.
(ii) log
_{3} (1/3)
Solution:
Let log
_{3} (1/3) = x
⇒ 3
^{x} = 1/3
⇒ 3
^{x} = 3
^{1}
⇒ x = 1,
Therefore, log
_{3}(1/3) = 1.
(iii) log
_{5} 0.008
Solution:
Let log
_{5} 0.008 = x
⇒ 5
^{x} = 0.008
⇒ 5
^{x} = 1/125
⇒ 5
^{x} = 5
^{3}
⇒ x = 3,
Therefore, log
_{5} 0.008 = 3.
4. Solve the following for x:
(i) log
_{x} 243 = 5
Solution:
log
_{x} 243 = 5
⇒ x
^{5} = 243
⇒ x
^{5} = 3
^{5}
⇒ x
^{5} = (1/3)
^{5}
⇒ x = 1/3.
(ii) log
_{√5} x = 4
Solution:
log
_{√5} x = 4
⇒ x = (√5)
^{4}
⇒ x = (5
^{1/2})
^{4}
⇒ x = 5
^{2}
⇒ x = 25.
(iii) log
_{√x} 8 = 6
Solution:
log
_{√x} 8 = 6
⇒ (√x)
^{6} = 8
⇒ (x
^{1/2})
^{6} = 2
^{3}
⇒ x
^{3} = 2
^{3}
⇒ x = 2.
Logarithmic Form Vs. Exponential Form
The logarithm function with base a has domain all positive real numbers and is defined by
log_{a} M = x ⇔ M = a^{x}
where M > 0, a > 0, a ≠ 1
Logarithmic Form Exponential Form
log_{a} M = x ⇔ M = a^{x}
Log_{7} 49 = 2 ⇔ 7^{2} = 49
● Write the exponential equation in logarithmic form.
Exponential Form Logarithmic Form
M = a^{x} ⇔ log_{a} M = x
2^{4} = 16 ⇔ log_{2} 16 = 4
10^{2} = 0.01 ⇔ log_{10} 0.01 = 2
8^{1/3} = 2 ⇔ log_{8} 2 = 1/3
6^{1} = 1/6 ⇔ log_{6} 1/6 = 1
● Write the logarithmic equation in exponential form.
Logarithmic Form Exponential Form
log_{a} M = x ⇔ M = a^{x}
log_{2} 64 = 6 ⇔ 2^{6} = 64
log_{4} 32 = 5/2 ⇔ 4^{5/2}= 32
log_{1/8}2 = 1/3 ⇔ (1/8)^{1/3} = 2
log_{3} 81 = x ⇔ 3^{x} = 81
log_{5} x = 2 ⇔ 5^{2} = x
log x = 3 ⇔ 10^{3} = x
● Solve for x:
1. log_{5} x = 2
x = 5
^{2}
= 25
2. log_{81} x = ½
x = 81
^{1/2}
⇒ x= (9
^{2})
^{1/2}
⇒ x = 9
3. log_{9} x = 1/2
x = 9
^{1/2}
⇒ x = (3
^{2})
^{1/2}
⇒ x = 3
^{1}
⇒ x= 1/3
4. log_{7} x = 0
x= 7
^{0}
⇒ x = 1
● Solve for n:
1. log_{3} 27 = n
3
^{n} = 27
⇒ 3
^{n} = 3
^{3}
⇒ n = 3
2. log_{10} 10,000 = n
10
^{n} = 10,000
⇒ 10
^{n} = 10
^{4}
⇒ n = 4
3. log_{49} 1/7 = n
49
^{n} = 1/7
⇒ (7
^{2})
^{n} = 7
^{1}
⇒ 7
^{2n} = 7
^{1}
⇒ 2n = 1
⇒ n = 1/2
4. log_{36} 216 = n
36
^{n} = 216
⇒ (6
^{2})
^{n} = 6
^{3}
⇒ 6
^{2n}= 6
^{3}
⇒ 2n = 3
⇒ n = 3/2
● Solve for b:
1. log_{b} 27 = 3
b
^{3} = 27
⇒ b
^{3} = 3
^{3}
⇒ b = 3
2. log_{b} 4 = 1/2
b
^{1/2} = 4
⇒ (b
^{1/2})
^{2} = 4
^{2}
⇒ b = 16
3. log_{b} 8 = 3
b
^{3} = 8 ⇒ b
^{3} = 2
^{3}
⇒ (b
^{1})
^{3} = 2
^{3}
⇒b
^{1} = 2
⇒ 1/b = 2
⇒ b = ½
4. log_{b} 49 = 2
b
^{2} = 49
⇒ b
^{2} = 7
^{2}
⇒ b = 7
● If f(x) = log_{3} x, find f(1).
Solution:
f(1) = log
_{3} 1 = 0 (since logarithm of 1 to any finite nonzero base is zero.)
Therefore f(1) = 0
● A number that is domain of the function y = log_{10} x is
(a) 1
(b) 0
(c) ½
(d) =10
Answer: (b)
● The graph of y = log_{4} x lines entirely in quadrants
(a) I and II
(b) II and III
(c) I and III
(d) I and IV
● At what point does the graph of y = log_{5} x intersect the xaxis?
(a) (1, 0)
(b) (0, 1)
(c) (5, 0)
(d) There is no point of intersection.
Answer: (a)
● Mathematics Logarithm
Mathematics Logarithms
Convert Exponentials and Logarithms
Logarithm Rules or Log Rules
Solved Problems on Logarithm
Common Logarithm and Natural Logarithm
Antilogarithm
Logarithms
11 and 12 Grade Math
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