Selina solutions for Concise Mathematics [English] Class 9 ICSE chapter 8 - Logarithms [Latest edition]

Express the following in logarithmic form : 5³ = 125

Express the following in logarithmic form :
3^{-2 =} `1/9`

Express the following in logarithmic form :
10^-3 = 0.001

Express the following in logarithmic form : `(81)^(3/4) = 27` 

Express the following in exponential form : log₈0.125 = -1

Express the following in exponential form : 
log₁₀0.01 = - 2

Express the following in exponential form : log_aA = x

Express the following in exponential form : log₁₀1 = 0

Solve for x : log₁₀ x = -2.

Find the logarithm of : 100 to the base 10

Find the logarithm of : 0.1 to the base 10

Find the logarithm of : 0.001 to the base 10

Find the logarithm of : 32 to the base 4

Find the logarithm of : 0.125 to the base 2

Find the logarithm of : `1/16` to the base 4

Find the logarithm of : 27 to the base 9

Find the logarithm of : `1/81` to the base 27

State, true or false : If log₁₀ x = a, then 10^x = a.

State, true or false : If x^y = z, then y = log_zx .

State, true or false : log₂ 8 = 3 and log₈ = 2 = `1/3`

Find x, if : log₃ x = 0

Find x, if : log_x 2 = - 1.

Find x, if : log₉243 = x

Find x, if : log₅ (x - 7) = 1

Find x, if : log₄32 = x - 4

Find x, if : log₇ (2x² - 1) = 2

Evaluate : log₁₀ 0.01

Evaluate : log₂ ( 1 ÷ 8 )

Evaluate : log₅ 1

Evaluate : log₅ 125

Evaluate : log₁₆ 8

Evaluate : log_0.5 16

If log_a m = n, express a^{n - 1} in terms of a and m.

Given log₂ x = m. Express 2^{m - 3}  in terms of x.

Given log₅ y = n. Express 5^{3n + 2} in terms of y.

If log₂x = a  and log₃ y = a, write 72^a in terms of x and y.

Solve for x:

log(x - 1) + log (x + 1) = log₂1

If log (x² - 21) = 2, show that x = ± 11.

Express in terms of log 2 and log 3 : log 36

Express in terms of log 2 and log 3:

log 144

Express in terms of log 2 and log 3 : log 4.5

Express in terms of log 2 and log 3 :
`"log"26/51 - "log"91/119`

Express in terms of log 2 and log 3 :
`"log"75/16 - 2"log"5/9 + "log"32/243`

Express the following in a form free from logarithm:

2 log x - log y = 1

Express the following in a form free from logarithm:
2 log x + 3 log y = log a

Express the following in a form free from logarithm:
a log x - b log y = 2 log 3

Evaluate  the following without using tables : 
log 5 + log 8 - 2 log 2

Evaluate the following without using tables :
log 4 + `1/3` log 125 - `1/5`log 32 

Evaluate the following without using tables : 
log₁₀8 + log₁₀25 + 2 log₁₀3 - log₁₀18

Prove that : `2"log" 15/18 - "log"25/162 + "log"4/9 = log 2 `

Find x, if : x - log 48 + 3 log 2 = `1/3`log 125 - log 3.

Express log₁₀2 + 1 in the form of log₁₀x .

Solve for x : log₁₀ (x - 10) = 1

Solve for x : log (x² - 21) = 2.

Solve for x :  log (x - 2) + log (x + 2) = log 5

Solve for x : log (x + 5) + log (x - 5) = 4 log 2 + 2 log 3

Solve for x :  `(log 81)/(log27 )` = x 

Solve for x : ` ( log 128) / ( log 32 ) ` = x

Solve for x : ` (log 64)/(log 8)` = log x

Solve for x :
`log 225/log15` = log x

Given that log x = m + n and log y = m - n, express the value of log ` ( 10x  ) / ( y ^ 2  )`  in terms of m and n.

State, true or false : log 1 x log 1000 = 0

State, true or false : 
`log x/log y` = log x - log y

State, true or false : 
If `log 25/log 5 = log x`, then x = 2.

State, true or false : 
log x x log y = log x + log y

If log₁₀2 = a and log₁₀3 = b ; express each of the following in terms of 'a' and 'b': log 12

If log₁₀2 = a and log₁₀3 = b ; express each of the following in terms of 'a' and 'b': log 2.25

If log₁₀2 = a and log₁₀3 = b ; express each of the following in terms of 'a' and 'b': log `2 1/4`

If log₁₀2 = a and log₁₀3 = b ; express each of the following in terms of 'a' and 'b' : log 5.4 

If log₁₀2 = a and log₁₀3 = b; express each of the following in terms of 'a' and 'b' : log 60

If log₁₀2 = a and log₁₀3 = b; express each of the following in terms of 'a' and 'b' :  log `3 1/8`

If log 2 = 0.3010 and log 3 = 0.4771 ;  find the value of : log 12

If log 2 = 0.3010 and log 3 = 0.4771 ; find the value of : log 1.2

If log 2 = 0.3010 and log 3 = 0.4771; find the value of : log 3.6

If log 2 = 0.3010 and log 3 = 0.4771; find the value of : log 15

If log 2 = 0.3010 and log 3 = 0.4771; find the value of : log 25

If log 2 = 0.3010 and log 3 = 0.4771; find the value of:

`2/3` log 8

Given 2 log₁₀ x + 1 = log₁₀ 250, find :
(i) x
(ii) log₁₀ 2x

Given 3log x + `1/2`log y = 2, express y in term of x.

If x = (100)^a , y = (10000)^b and z = (10)^c , find log`(10sqrty)/( x^2z^3)` in terms of a, b and c.

If 3( log 5 - log 3 ) - ( log 5 - 2 log 6 ) = 2 - log x, find x.

If log₁₀ 8 = 0.90; find the value of : log₁₀ 4

If log₁₀ 8 = 0.90; find the value of : log√32

If log₁₀ 8 = 0.90; find the value of : log 0.125

If log 27 = 1.431, find the value of : log 9

If log 27 = 1.431, find the value of : log 300

If log₁₀ a = b, find 10^{3b - 2} in terms of a.

If log₅ x = y, find 5^{2y+ 3} in terms of x.

Given: log₃ m = x and log₃ n = y.
Express 3^{2x - 3} in terms of m.

Given: log₃ m = x and log₃ n = y.

Write down `3^(1 - 2y + 3x)` in terms of m and n.

Given: log₃ m = x and log₃ n = y.
If 2 log₃ A = 5x - 3y; find A in terms of m and n.

Simplify : log (a)³ - log a

Simplify : log (a)³ ÷  log a

If log (a + b) = log a + log b, find a in terms of b.

Prove that :  (log a)² - (log b)² = log `(( a )/( b ))` . Log (ab)

Prove that :  If a log b + b log a - 1 = 0, then b^a. a^b = 10

 If log (a + 1) = log (4a - 3) - log 3; find a.

If 2 log y - log x - 3 = 0, express x in terms of y.

Prove that:

log₁₀ 125 = 3(1 - log₁₀2).

Given log x = 2m - n , log y = n - 2m and log z = 3m - 2n , find in terms of m and n, the value of log `(x^2y^3 ) /(z^4) `.

Given `log_x 25 - log_x 5 = 2 - log_x (1/125)` ; find x.

If `3/2 log a + 2/3` log b - 1 = 0, find the value of a⁹.b⁴ .

If x = 1 + log 2 - log 5, y = 2 log3 and z = log a - log 5; find the value of a if x + y = 2z.

If x = log 0.6; y = log 1.25 and z = log 3 - 2 log 2, find the values of :
(i) x+y- z       
(ii) 5^{x + y - z}

If a² = log x, b³ = log y and 3a² - 2b³ = 6 log z, express y in terms of x and z .

If log`( a - b )/2 = 1/2( log a + log b )`, Show that : a² + b² = 6ab. 

If a² + b² = 23ab, show that:

log `(a + b)/5 = 1/2`(log a + log b).

If m = log 20 and n = log 25, find the value of x, so that :
2 log (x - 4) = 2 m - n.

Solve for x and y ; if x > 0 and y > 0 ; log xy = log `x/y` + 2 log 2 = 2.

Find x, if :  log_x 625 = - 4

Find x, if : log_x (5x - 6) = 2

Evaluate :  `( log _5^8 )/(( log_25 16 ) xx  ( log_100  10))`

If p = log 20 and q = log 25 , find the value of x , if 2log( x + 1 ) = 2p - q. 

If log₂(x + y) = log₃(x - y) = `log 25/log 0.2`, find the values of x and y.

Given : `log x/ log y = 3/2` and log (xy) = 5; find the value of x and y.

Given log₁₀x = 2a and log₁₀y = `b/2`. Write 10^a in terms of x.

Given log₁₀x = 2a and log₁₀y = `b/2`. Write 10^{2b + 1} in terms of y.

Given log₁₀x = 2a and log₁₀y = `b/2. "If"  log_10^p = 3a - 2b`, express P in terms of x and y.

Solve : log₅( x + 1 ) - 1 = 1 + log₅( x - 1 ).

Solve for x, if : log_x49 - log_x7 + log_x `1/343` + 2 = 0

If a² = log x , b³ = log y and `a^2/2 - b^3/3` = log c , find c in terms of x and y.

Given x = log₁₀12 , y = log₄ 2 x log₁₀9 and z = log₁₀0.4 , find :
(i) x - y - z
(ii) 13^{x - y - z}

Solve for x, `log_x^(15√5) = 2 - log_x^(3√5)`.

Evaluate: log_b a × log_c b × log_a c.

Evaluate :  log₃8 ÷ log₉16 

Evaluate: `(log_5 8)/(log_25 16 xx Log_100 10)` 

Show that : log_a m ÷ log_ab m + 1 + log _ab 

If log_√27x = 2 `(2)/(3)` , find x.

Evaluate :`1/( log_a bc + 1) + 1/(log_b ca + 1) + 1/ ( log_c ab + 1 )`