Logarithms Concepts

1. Basic Logarithm Properties & Simplification

• Formula:

$$\log_a(MN)=\log_aM+\log_aN$$ $$\log_a\left(\frac MN\right)=\log_aM-\log_aN$$ $$\log_a(M^n)=n\log_aM$$ $$\log_a a=1,\qquad \log_a1=0$$

Use these to expand or combine logarithmic expressions.

• Example: Simplify (\log(8\times125)). Solution:

$$\log(8\times125)=\log8+\log125$$ $$=\log2^3+\log5^3$$ $$=3\log2+3\log5$$ $$=3\log(10)$$ $$\boxed{3}$$

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2. Combining Logarithms

• Formula:

$$\log_aM+\log_aN=\log_a(MN)$$ $$\log_aM-\log_aN=\log_a\left(\frac MN\right)$$

Use when several logarithms with the same base need to be combined.

• Example: Simplify

$$\log_2 12-\log_2 3+\log_2 4.$$

Solution:

$$=\log_2\left(\frac{12\times4}{3}\right)$$ $$=\log_2 16$$ $$\boxed{4}$$

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3. Change of Base

• Formula:

$$\boxed{\log_a b=\frac{\log_c b}{\log_c a}}$$

In particular:

$$\log_a b=\frac{\ln b}{\ln a}$$

Use when the required base is inconvenient or different logarithms have different bases.

• Example: Find (\log_2 8) using common logarithms. Solution:

$$\log_2 8=\frac{\log10 8}{\log10 2}$$

Since:

$$8=2^3$$ $$\log_2 8=\boxed{3}$$

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4. Converting Between Logarithmic and Exponential Forms

• Formula:

$$\boxed{\log_a x=y\iff a^y=x}$$

where (a>0,\ a\neq1,\ x>0).

• Example: Solve

$$\log_3x=4.$$

Solution:

Convert to exponential form:

$$x=3^4$$ $$\boxed{x=81}$$

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5. Basic Logarithmic Equations

• Formula:

$$\log_a f(x)=\log_a g(x) \Rightarrow f(x)=g(x)$$

provided both arguments are positive.

• Example: Solve

$$\log_2(x+3)=\log_2(2x-1).$$

Solution:

Equate the arguments:

$$x+3=2x-1$$ $$x=4$$

Check:

$$x+3=7>0,\qquad2x-1=7>0$$

Therefore:

$$\boxed{x=4}$$

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6. Logarithmic Equations with Different Bases

• Formula:

$$\log_a x=\frac{\log_bx}{\log_ba}$$

Convert all logarithms to a common base.

• Example: Solve

$$\log_2x+\log_4x=6.$$

Solution:

Let:

$$y=\log_2x$$

Since:

$$\log_4x=\frac{\log_2x}{\log_24} =\frac y2$$

Therefore:

$$y+\frac y2=6$$ $$\frac{3y}{2}=6$$ $$y=4$$

Hence:

$$x=2^4$$ $$\boxed{x=16}$$

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7. Logarithmic Equations Using Product/Quotient Rules

• Formula:

$$\log_aM+\log_aN=\log_a(MN)$$ $$\log_aM-\log_aN=\log_a(M/N)$$

Combine first, then convert to exponential form.

• Example: Solve

$$\log_3x+\log_3(x-2)=1.$$

Solution:

Domain:

$$x>0,\quad x-2>0$$

Thus:

$$x>2$$

Combine:

$$\log_3[x(x-2)]=1$$

Therefore:

$$x(x-2)=3$$ $$x^2-2x-3=0$$ $$(x-3)(x+1)=0$$ $$x=3,-1$$

Only (x=3) satisfies (x>2).

$$\boxed{x=3}$$

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8. Characteristic and Mantissa

• Formula:

$$\log_{10}N=\text{Characteristic}+\text{Mantissa}$$

For (N>1):

$$\boxed{\text{Characteristic}=\text{number of digits}-1}$$

If (0<N<1), the characteristic is negative and is handled using bar notation.

• Example: Given (\log10 2=0.3010), find the number of digits in (2^{50}). Solution:

$$\log(2^{50})=50\log2$$ $$=50(0.3010)=15.05$$

Therefore:

$$\text{Characteristic}=15$$

Number of digits:

$$15+1=\boxed{16}$$

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9. Number of Digits Using Logarithms

• Formula: If (N>0) and (N) is not an exact power of 10:

$$\boxed{\text{Number of digits}=\lfloor\log_{10}N\rfloor+1}$$

For (a^n):

$$\boxed{\text{digits}=\lfloor n\log_{10}a\rfloor+1}$$

• Example: How many digits are there in (3^{20}), given (\log3=0.4771)? Solution:

$$\log(3^{20})=20(0.4771)=9.542$$

Therefore:

$$\text{digits}=\lfloor9.542\rfloor+1$$ $$=9+1$$ $$\boxed{10}$$

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10. Logarithmic Inequalities — Base Greater Than 1

• Formula: If (a>1):

$$\boxed{\log_a x>\log_a y\iff x>y}$$

If:

$$\log_a f(x)>k$$

then:

$$f(x)>a^k$$

Always impose (f(x)>0).

• Example: Solve

$$\log_2(x-1)>3.$$

Solution:

Since (2>1):

$$x-1>2^3$$ $$x-1>8$$ $$\boxed{x>9}$$

The domain (x>1) is automatically satisfied.

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11. Logarithmic Inequalities — Base Between 0 and 1

• Formula: If:

$$0<a<1,$$

the inequality reverses:

$$\boxed{\log_a x>\log_a y\iff x<y}$$

Always require (x>0).

• Example: Solve

$$\log_{0.5}(x-1)>-2.$$

Solution:

Since:

$$0<0.5<1,$$

the inequality reverses:

$$x-1<(0.5)^{-2}$$ $$x-1<4$$ $$x<5$$

Domain:

$$x-1>0\Rightarrow x>1$$

Therefore:

$$\boxed{1<x<5}$$

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12. Domain Restrictions in Logarithms

• Formula: For every logarithm:

$$\boxed{\text{argument}>0}$$

and:

$$\boxed{\text{base}>0,\quad\text{base}\neq1}$$

Apply these conditions after solving the equation.

• Example: Solve

$$\log_2(x^2-9)=2.$$

Solution:

Convert to exponential form:

$$x^2-9=2^2$$ $$x^2=13$$ $$x=\pm\sqrt{13}$$

Both values give:

$$x^2-9=4>0$$

Therefore:

$$\boxed{x=\pm\sqrt{13}}$$

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13. Logarithm of a Power

• Formula:

$$\boxed{\log_a(x^n)=n\log_ax}$$

Also:

$$\boxed{\log_{a^m}x=\frac1m\log_ax}$$

Use to reduce powers or convert between bases.

• Example: Simplify

$$\log_2(32^3).$$

Solution:

$$32=2^5$$

Therefore:

$$\log_2(32^3) ============ \log_2(2^{15})$$ $$\boxed{15}$$

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14. Reciprocal and Negative Logarithms

• Formula:

$$\boxed{\log_a\frac1x=-\log_ax}$$ $$\boxed{\log_{1/a}x=-\log_ax}$$

Use when reciprocals or reciprocal bases appear.

• Example: Simplify

$$\log_2\frac18.$$

Solution:

$$\frac18=2^{-3}$$

Therefore:

$$\log_2(2^{-3})=-3$$ $$\boxed{-3}$$

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15. Exponential Equations Using Logarithms

• Formula: For:

$$a^x=b,$$ $$\boxed{x=\frac{\log b}{\log a}}$$

Use when the equation cannot be solved conveniently by matching powers.

• Example: Solve

$$5^x=20.$$

Solution:

Take logarithms:

$$x\log5=\log20$$

Therefore:

$$x=\frac{\log20}{\log5}$$ $$\boxed{x=\log_5 20}$$

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Advanced Variants

16. Nested Logarithms

• Formula:

$$\boxed{\log_a(\log_bx)=c \Rightarrow \log_bx=a^c \Rightarrow x=b^{a^c}}$$

Domain restrictions must be checked at every level.

• Example: Solve

$$\log_2(\log_3x)=2.$$

Solution:

$$\log_3x=2^2=4$$

Therefore:

$$x=3^4$$ $$\boxed{x=81}$$

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17. Logarithmic Equations with Parameter

• Formula: Combine logarithms first, convert to an algebraic equation, then apply the logarithm domain restrictions.
• Example: Solve

$$\log_2(x-1)+\log_2(x+1)=3.$$

Solution:

Domain:

$$x>1$$

Combine:

$$\log_2[(x-1)(x+1)]=3$$ $$x^2-1=8$$ $$x^2=9$$ $$x=\pm3$$

Since (x>1):

$$\boxed{x=3}$$

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18. Logarithmic Inequality with Multiple Terms

• Formula: Combine logarithms using:

$$\log_aM+\log_aN=\log_a(MN)$$

and remember that the inequality reverses when (0<a<1).

• Example: Solve

$$\log_2x+\log_2(x-2)>3.$$

Solution:

Domain:

$$x>2$$

Combine:

$$\log_2[x(x-2)]>3$$

Since (2>1):

$$x(x-2)>8$$ $$x^2-2x-8>0$$ $$(x-4)(x+2)>0$$

Thus:

$$x<-2\quad\text{or}\quad x>4$$

Applying (x>2):

$$\boxed{x>4}$$

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19. Change-of-Base Product/Cyclic Form

• Formula:

$$\boxed{\log_ab\cdot\log_bc=\log_ac}$$

and:

$$\boxed{\log_ab\cdot\log_bc\cdot\log_ca=1}$$

Useful for aptitude questions involving cyclic logarithms.

• Example: Evaluate

$$\log_2 3\cdot\log_3 5\cdot\log_5 8.$$

Solution:

Using:

$$\log_ab\cdot\log_bc=\log_ac$$ $$\log_2 3\cdot\log_3 5=\log_2 5$$

Therefore:

$$\log_2 5\cdot\log_5 8 =\log_2 8$$ $$\boxed{3}$$

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20. Base Conditions in Logarithmic Inequalities

• Formula: For:

$$\log_a x$$

if (a>1), the function is increasing; if (0<a<1), it is decreasing.

$$\boxed{a>1:\text{ preserve inequality}}$$ $$\boxed{0<a<1:\text{ reverse inequality}}$$

• Example: Solve

$$\log_{1/3}(2x-1)\leq2.$$

Solution:

Since:

$$0<\frac13<1,$$

reverse the inequality:

$$2x-1\geq\left(\frac13\right)^2$$ $$2x-1\geq\frac19$$ $$2x\geq\frac{10}{9}$$ $$x\geq\frac59$$

Domain:

$$2x-1>0\Rightarrow x>\frac12$$

Combining:

$$\boxed{x\geq\frac59}$$