
Math`log_4(128)` To evaluate, factor 128. `= log_4 (2^7)` Then, apply the formula of change base `log_b (a) = (log_c (a))/(log_c (b))` . `= (log_2 (2^7))/(log_2 (4))` `= (log_2 (2^7))/(log_2 (2^2))` To...

Math`log_125 (625)` To evaluate, factor 625. `=log_125 (5^4)` Then, apply the formula of change base `log_b(a) = (log_c(a))/(log_c(b))` . `= (log_5 (5^4))/(log_5 (125))` `= (log_5 (5^4))/(log_5...

Math`log_8 (32)` To evaluate, factor 32. `=log_8 (2^5)` Then, apply the formula of change base `log_b(a) = (log_c(a))/(log_c(b))` . `= (log_2 (2^5))/(log_2 (8))` `=(log_2(2^5))/(log_2(2^3))` Also,...

Math`log_27 (9)` To evaluate, factor the 9. `= log_27 (3^2)` Then, apply the formula of change base `log_b (a) =(log_c(a))/(log_c(b))` . `= (log_3(3^2))/(log_3 (27))` `=(log_3 (3^2))/(log_3 (3^3))` To...

MathWe are asked to graph the following function: `y=log_5(x+1)3` The base function is `y=log_5(x)` and the graph will be a translation of 1 unit left and 3 units down of the graph of the base...

MathWe are asked to graph the function `y=log_6(x4)+2 ` : The graph is a translation of the graph of `y=log_6x ` 4 units right and 2 units up. The domain is x>4 and the range is all real numbers....

MathWe are asked to graph the function `y=log_4(x+2)1 ` : This is a translation of the graph of ` y=log_4x ` 2units left and 1 unit down. The domain is x>2 and the range is all real numbers. There...

MathWe are asked to graph the function `y=log_3x+4 ` : The graph is a translation of the graph `y=log_3x ` up 4 units. Some points on the graph: (1/27,1),(1/9,2),(1/3,3),(1,4),(3,5),(9,6) The domain is...

MathWe are asked to graph the function `y=log_2(x3) ` : The graph is a translation of the graph of `y=log_2x ` 3 units right. Some points on the graph:...

MathWe are asked to graph the following function: `y=log_(1/5)(x)` The domain is x>0 and the range is all real numbers. The graph is decreasing and concave up on the domain. Some points on the...

MathWe are asked to graph the following function: `y=log_(1/3)(x)` The domain is x>0 and the range is all real numbers. The graph of the function is decreasing and concave up on its domain. Some...

MathWe are asked to graph the following function: `y=log_6(x)` The domain is x>0 and the range is all real numbers. The graph is increasing on the domain, and the graph is concave down on the...

MathWe are asked to graph the following function: `y=log_4(x)` The domain is x>0 and the range is all real numbers. The graph increases on its domain and is concave down on its domain. Some points...

Math`2(log_3 (20)  log_3 (4)) + 0.5log_3(4)` First, apply the differencequotient rule of logarithm `log_b (m/n) = log_b(m)  log_b(n)` . `= 2 (log_3 (20/4))+0.5log_3(4)` `=2log_3(5) + 0.5log_3(4)`...

Math`ln40+2ln(1/2) + lnx` First, apply the logarithm rule `log_b (a^m)=m*log_b(a)` . `= ln40 + ln(1/2)^2 + ln x` `=ln40+ ln(1/4) + lnx` And, apply the rule `log_b(m*n) = log_bm +log_b n` . `= ln (40 *...

Math`5log_4(2) + 7log_4(x) + 4log_4(y)` To express this as one logarithm, first apply the rule `log_b a^m = m*log_b(a)` . `= log_4(2^5) + log_4(x^7) + log_4(y^4)` `= log_4(32) + log_4(x^7)+log_4(y^4)`...

MathWe are asked to solve `5^(2x)+20*5^x125=0 ` : Rewrite as ` (5^x)^2+20*5^x125=0 ` and let `y=5^x ` to get ` y^2+20y125=0` and (y+25)(y5)=0 so y=25 or y=5. y cannot be 25 as `5^x>0 ` for all...

MathWe are asked to solve `2^(2x)12*2^x+32=0 ` : Rewrite as `(2^x)^212*2^x+32=0 ` and let `y=2^x ` ; then `y^212y+32=0 ` and (y8)(y4)=0 so y=8 or y=4. y=8 ==> ` 2^x=8 ==> x=3 ` y=4 ==>...

MathTo solve the equation log_3(x)=log_9(6x), we may apply logarithm properties. Apply the logarithm property: `log_a(b)= (log_c(b))/log_c(a)` on `log_3(x)` , we get: `(log_9(x))/(log_9(3))=log_9(6x)`...

MathSolve `log_2(x+1)=log_8(3x) ` : Rewrite using the change of base formula: `(ln(x+1))/(ln(2))=(ln(3x))/(ln(8)) ` `(ln(8))/(ln(2))=(ln(3x))/(ln(x+1)) ` But `ln(8)=ln(2^3)=3ln(2) ` so:...

MathTo solve the equation: `10^(3x8)=2^(5x)` , we may take "ln" on both sides. `ln(10^(3x8))=ln(2^(5x))` Apply natural logarithm property: `ln(x^n) = n*ln(x)` . `(3x8)ln(10)=(5x)ln(2)` Let...

MathRewrite `6^(2x5)' as `(2 times 3)^(2x5) = 3^(2x5)2^(2x5)' Divide both sides of the equation through by `3^(x+4)' giving `1 = 3^(2x5x4)2^(2x5)' That is, `3^(x9)2^(2x5) = 1' Take logs (with...

MathTo evaluate the given equation `log_6(3x)+log_6(x1)=3` , we may apply the logarithm property: `log_b(x)+log_b(y)=log_b(x*y)` . `log_6(3x)+log_6(x1)=3` `log_6(3x*(x1))=3` `log_6(3x^23x)=3` To...

MathWe are asked to solve the following equation: `4ln(x)+3=21` Use basic algebraic rules to isolate the term with the logarithm: `4ln(x)=18` `ln(x)=9/2` Exponentiate both sides with base e:...

MathTo evaluate the given equation `log_4(x)+log_4(x+10)=2` , we may apply the logarithm property: `log_b(x)+log_b(y)=log_b(x*y)` . `log_4(x)+log_4(x+10)=2` `log_4((x)*(x+10))=2`...

MathTo evaluate the given equation `log_2(x4)=6` , we may apply the logarithm property: `a^(log_a(x))=x` . Raised both sides by base of `2` . `2^(log_2(x4))=2^6` `x4=64` Add `4` on both sides to...

MathTo evaluate the given equation `1/3log_5(12x)=2` , we may apply logarithm property: `n* log_b(x) = log_b(x^n)` . `log_5((12x)^(1/3))=2` Take the "log" on both sides to be able to apply the...

MathWe are asked to solve `log_8(512x)=log_8(6x1)` Exponentiating both sides with base 8 we get: 512x=6x118x=6 x=1/3. This is in the domain of both expressions of the equality (the domain for the...

MathTo evaluate the equation `log_6(3x10)=log_6(145x)` , we apply logarithm property: `a^(log_a(x))=x` . Raised both sides by base of `6` . `6^(log_6(3x10))=6^(log_6(145x))` `3x10=145x` Add `10`...

MathTo solve the equation `log_3(18x+7)=log_3(3x+38)` , we apply logarithm property: `a^(log_a(x))=x` . Raised both sides by base of `3` . `3^(log_3(18x+7))=3^(log_3(3x+38))` `18x+7=3x+38` Subtract 7...

Math`log(12x11)=log(3x+13)` Using the property of logarithmic equality, `12x11=3x+13` `=>12x3x=13+11` `=>9x=24` `=>x=24/9` `=>x=8/3` Let's plug back the solution in the equation to check...

Math`log_5(2x7)=log_5(3x9)` Using one to one property of logarithms, `2x7=3x9` `=>2x3x=9+7` `=>x=2` `=>x=2` Let's plug back the solution in the equation, `log_5(2*27)=log_5(3*29)`...

MathTo evaluate the equation `ln(x+19)=ln(7x8)` , we apply natural logarithm property: `e^(ln(x))=x` . Raise both sides by base of `e` . `e^(ln(x+19))=e^(ln(7x8))` `x+19=7x8` Subtract `7x` from both...

Math`ln(4x7)=ln(x+11)` Using one to one property of logarithms, `4x7=x+11` `=>4xx=11+7` `=>3x=18` `=>x=18/3` `=>x=6` Plug back the solution in the equation to check the solution,...

MathTo solve the equation `log_5(5x+9)=log_5(6x)` , we apply logarithm property: `a^(log_a(x))=x` . Raise both sides by base of `5` . `5^( log_5(5x+9))=5^(log_5(6x))` `5x+9=6x` Subtract `5x` from...

MathFor the given equation `2^(0.1x)5=12` , we may simplify by combining like terms. Add `5` on both sides of the equation. `2^(0.1x)5+5=12+5` `2^(0.1x)=17` Take the "`ln` " on both sides to be able...

MathFor the given equation `0.5^x0.25=4` , we may simplify by combining like terms. Add `0.25` on both sides of the equation. `0.5^x0.25+0. 25=4+0.25` `0.5^x=4.25` Take the "`ln` " on both sides to...

MathFor the given equation `10^(3x)+4 =9` , we may simplify by combining like terms. Subtract 4 from both sides of the equation. `10^(3x)+44 =94` `10^(3x)=5` Take the "ln" on both sides to be able to...

MathTo solve the given equation `7^(6x)=12` , we may take "`ln` " on both sides of the equation. `ln(7^(6x))=ln(12)` Apply natural logarithm property: `n*ln (x)=ln (x^n)` . `6x*ln(7)=ln(12)` Divide...

MathTo solve the given equation `7^(3x)=18` , we may take "ln" on both sides of the equation. `ln(7^(3x))=ln(18)` Apply natural logarithm property:` ln (x^n) = n*ln (x)` . `3xln(7)=ln(18)` Divide both...

MathTo solve the given equation `8^x=20` , we may take "ln" on both sides of the equation. `ln(8^x)=ln(20)` Apply natural logarithm property: `ln (x^n) = n*ln (x)` . `xln(8)=ln(20)` Divide both sides...

MathTo evaluate the given equation `25^(10x+8)=(1/125)^(42x)` , we may apply `25=5^2` and `1/125=5^(3)` . The equation becomes: `(5^2)^(10x+8)=(5^(3))^(42x)` Apply Law of Exponents: `(x^n)^m =...

MathTo evaluate the given equation `10^(3x10)=(1/100)^(6x1)` , we may apply `100=10^2` . The equation becomes: `10^(3x10)=(1/10^2)^(6x1)` Apply Law of Exponents: `1/x^n = x^(n)` ....

MathTo evaluate the given equation `36^(5x+2)=(1/6)^(11x)` , we may apply `36=6^2` and `1/6=6^(1)` . The equation becomes: `(6^2)^(5x+2)=(6^(1))^(11x)` Apply Law of Exponents: `(x^n)^m = x^(n*m)`...

Math`3^(3x7)=81^(123x)` To solve, factor 81. `3^(3x7)=(3^4)^(123x)` To simplify the right side, apply the exponent rule `(a^m)^n=a^(m*n)` . `3^(3x7)=3^(4*(123x))` `3^(3x7)= 3^(4812x)` Since...

MathTo evaluate the given equation `4^(2x5)=64^(3x)` , we may let `64 =4^3` . The equation becomes: `4^(2x5)=(4^3)^(3x)` . Apply Law of exponents: `(x^n)^m = x^(n*m)` . `4^(2x5)=4^(3*3x)`...

Math`27^(4x1)=9^(3x+8)` To solve, factor the 9 and 27. `(3^3)^(4x1)=(3^2)(^(3x+8)` To simplify each side, apply the exponent rule `(a^m)^n=a^(m*n)` . `3^(3*(4x1))=3^(2*(3x+8))`...

Math`8^(x1)=32^(3x2)` To solve, factor 8 and 32. `(2^3)^(x1)=(2^5)^(3x2)` To simplify each side, apply the exponent rule `(a^m)^n = a^(m*n)` . `2^(3*(x1)) = 2^(5*(3x2))` `2^(3x3) = 2^(15x10)`...

Math`7^(3x+4)=49^(2x+1)` To solve, factor the 49. `7^(3x+4)=(7^2)^(2x+1)` To simplify the right side, apply the exponent property `(a^m)^n=a^(m*n)` . `7^(3x+4)=7^(4x+2)` Since the two sides have the...

Math`5^(x4)=25^(x6)` To solve, factor the 25. `5^(x4)=(5^2)^(x6)` To simplify the right side, apply the exponent rule `(a^m)^n = a^(m*n)` . `5^(x4)=5^(2*(x6))` `5^(x4)=5^(2x12)` Since both...
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